I need a list of all of the European countries, their full country
code extension (i.e.  UK = mycompany.co.uk....I think), and most
importantly, how to register a URL in that country, the cost, and
whether there are restrictions on registration.  Certain countries
require citizenship, or property ownership, and I need to have this
information as well.

Hello B00ne

Thanks for the question, this subject is a complex one but hopefully
with my answer I will be able to cover all the information you
require.  Should you have any questions regarding this subject please
ask for clarification.

Regarding registration of these domain names the easiest way to
register them is through a domain name registrar who can often offer
you discounted prices if you are buying multiple domain names.  The
prices often vary from registrar to registrar and the prices I have
given here are given as a guide only.

The registrar I have used to uncover the information for this question
is located here: http://www.simply.com

Simply.com are very efficient and can even help you register domain
names if a local address is required in certain cases.  They can be
used to register all the domain extensions given in this answer.

ANDORRA
.ad domains are only available to organisations with an Andorran
registered trademark.  The annual price is £149.99 per year with a
minimum registration period of 1 year.

AUSTRIA
.at domains are available for anyone to register. There is no
requirement for residency or company ownership within Austria.  The
annual price is £89.99 per year with a minimum registration period of
1 year.

BELGIUM
.be domains are available for anyone to register. There is no
requirement for residency or company ownership within Belgium.  The
annual price is £29.99 per year with a minimum registration period of
1 year.

CZECH REPUBLIC
.cz domains are available for anyone to register. There is no
requirement for residency or company ownership within The Czech
Republic.  The annual price is £119.99 per year with a minimum
registration period of 1 year.

DENMARK
.dk domains are available for anyone to register. There is no
requirement for residency or company ownership within Denmark.  The
annual price is £29.99 per year with a minimum registration period of
1 year.

ESTONIA
.ee domains are only available to companies registered within Estonia.
 The annual price is £119.99 per year with a minimum registration
period of 1 year.

FINLAND
.fi domains are only available to associations in the Finnish Trade
Register, the Finnish Register of Associations or the Finnish Register
of Foundations.  The annual price is £99.99 per year with a minimum
registration period of 1 year.

FRANCE
.fr domains are only available to a French registered entity or
company with a SIREN/SIRET number or a holder of a European trademark.
 From the 11th May 2004 this was extended to companies or individuals
who can be identified on-line in official French national databases
(for example companies, the self-employed, associations listed in the
INSEE directory, trademark owners etc). The annual price is £79.99 per
year with a minimum registration period of 1 year.

GERMANY
.de domains are only available to applicants who have an
administration contact with a German address.  The annual price is
£29.99 per year with a minimum registration period of 1 year.

GIBRALTER
.gi domains are only available to Gibraltar based commercial
organisations.  The annual price is £149.99 per year with a minimum
registration period of 1 year.

GREECE
.gr domains are only available to available to Greek adults provided
they have a Greek VAT number, is a Greeks living abroad with proof of
nationality, or is a Greek or Foreign company with sufficient economic
activity in Greece.  The annual price is £99.99 per year with a
minimum registration period of 1 year.

GUERNSEY
.gg domains are available for anyone to register. There is no
requirement for residency or company ownership within Guernsey.  The
annual price is £149.99 per year with a minimum registration period of
1 year.

HUNGARY
.hu domains are only available to a Hungarian national person, a legal
entity or organisation or owners of a trademark registered by the
Hungarian Patent Office.  The annual price is £109.99 per year with a
minimum registration period of 1 year.

ISLE OF MANN
.co.im domains are available for anyone to register. There is no
requirement for residency or company ownership within The Isle of Man.
 The annual price is £119.99 per year with a minimum registration
period of 1 year.

ITALY
.it domains are available to any EU member citizen or EU member
registered company. Proof will be required of residency via Tax, VAT
or Company registration numbers. The annual price is £79.99 per year
with a minimum registration period of 1 year.

IRELAND
.ie domains are only available to only available to companies
incorporated under the laws of Ireland. Non-Irish companies with a
trading relationship with Ireland are allowed in certain
circumstances.  Irish citizens registering personal domains are also
allowed.  The annual price is £149.99 per year with a minimum
registration period of 1 year.

ISRAEL
.co.il domains are available for anyone to register. There is no
requirement for residency or company ownership within Israel.  The
annual price is £109.99 per year with a minimum registration period of
1 year.

JERSEY
.je domains are available for anyone to register. There is no
requirement for residency or company ownership within Jersey.  The
annual price is £149.99 per year with a minimum registration period of
1 year.

LIECHTENSTEIN
.li domains are available for anyone to register. There is no
requirement for residency or company ownership within Liechtenstein. 
The annual price is £89.99 per year with a minimum registration period
of 1 year.

LATVIA
.lv domains are available for anyone to register. There is no
requirement for residency or company ownership within Latvia.  The
annual price is £99.99 per year with a minimum registration period of
1 year.

LUXEMBOURG
.lu domains are available to those with a local contact and address. 
The annual price is £149.99 per year with a minimum registration
period of 1 year.

LITHUANIA
.lt domains are available for anyone to register.  The annual price is
£99.99 per year with a minimum registration period of 1 year.

MONACO
.mc domains are available for anyone with a local company and address.
 The annual price is £75 per year with a minimum registration period
of 1 year.

NETHERLANDS
.nl domains are available for anyone to register.  The annual price is
£29.99 per year with a minimum registration period of 1 year.

NORWAY
.no domains are only available to representatives or members of a
Norwegian registered organisation. The application must include the
organisation's registration number.  The annual price is £99.99 per
year with a minimum registration period of 1 year.

POLAND
.pl domains are available for anyone to register.  The annual price is
£119.99 per year with a minimum registration period of 1 year.

PORTUGAL
.pt domains are only available to applicants registered with the
Portuguese government. A registry number and a copy of their "Cartao
de Identificacao de Pessoa Colectiva" is required.  The annual price
is £119.99 per year with a minimum registration period of 1 year.

ROMANIA
.ro domains are available for anyone to register.  The annual price is
£99.99 per year with a minimum registration period of 1 year.

RUSSIA
.ru domains are available for anyone to register.  The annual price is
£99.99 per year with a minimum registration period of 1 year.

SLOVENIA
.si domains are only available for companies registered in Slovenia.
Domains can only be the full or abbreviated version of the company's
registered name.  The annual price is £119.99 per year with a minimum
registration period of 1 year.

SLOVAK REPUBLIC
 .sk domains are only available to companies with representation in
the Slovak Republic. The Domain can be used only in relation to
business in the Slovak Republic.  The annual price is £119.99 per year
with a minimum registration period of 1 year.

SPAIN
.es domains are only available to registrants with a NIF number, date
of company registration and OPEM inscription number.  The domain must
bear a direct relationship to company name.  The annual price is
£149.99 per year with a minimum registration period of 1 year.

SWEDEN
.se domains are available for anyone to register.  The annual price is
£39.99 per year with a minimum registration period of 1 year.

SWITZERLAND
.ch domains are available for anyone to register.  The annual price is
£89.99 per year with a minimum registration period of 1 year.

TURKEY
.com.tr domains are only available where proof of trademark
registration certificate can be produced for the domain you wish to
register.  The annual price is £99.99 per year with a minimum
registration period of 1 year.

UNITED KINGDOM
.co.uk domains are available for anyone to register.  The annual price
is £5.78 per year with a minimum registration period of 1 year.

Wow, that is a lot of information, I think I have covered the major
European countries and domains!  If you require any further help on
this subject please ask for clarification.

Excellent work, thanks for the great answers.  Very helpful, very complete.

In the UK, the subdomains .org.uk and .me.uk are also available.
co.uk, org.uk and me.uk are available at GBP 2.59 per year for two
years, + 17.5% vat, from www.123-reg.co.uk

Owain

There is also the proposed European Union TLD, .eu  This doesn't
appear to be operational yet though.

Owain

For those who may be interested in a fuller explanation of .uk domains
- There are in fact a few more sub categories in the .uk country code.

Nominet UK is the registry for .uk internet names. Nominet manage the
authoritative database of .uk domain name registrations.

This from their website:

"The sub categories are actually called Second Level Domains (SLDs).
People are encouraged to register a domain name within the SLD that
corresponds to their type of activity."

The SLDs are:

.co.uk for commercial enterprises (the largest SLD in the UK)

.me.uk for personal domains

.org.uk for non-commercial organisations

.ltd and .plc for registered company names only

.net.uk for Internet Service Providers

.sch.uk for schools 

There are a number of Second Level Domains (SLDs) that are not managed
by Nominet UK and are managed by other companies and organisations.
They are:

.ac.uk for Academic Establishments 
.gov.uk for Government Bodies 
.nhs.uk for NHS Organisations 
.police.uk for UK Police Forces 
.mod.uk for Ministry of Defence Establishments  


Source: 

http://www.nominet.org.uk/

As a clarification regarding .tr Turkish domain names. 

.com.tr domains cost $17usd/year  and requires either an
international/domestic patent registration or a registration from
chamber of commerce to register the matching domain name for a company
website.
.gen.tr costs less than $3usd/year  and doesn't have any prerequisites.