I am thinking of starting a business and would like to use the theme
of a specific cartoon to decorate it.  Do I need to check copy rights
or trade mark rights to do this if I'm displaying these things in
public? I'm thinking of using any kind of collectables that I can buy
like cartoon cells hanging on the walls, posters, life size models or
originals of characters, etc.  What about costumes either bought or
made by hand? What about creating give aways like mugs or small toys?
The name of the business will not contain the name of the cartoon.

Stuff you buy in with the images on, will ... probably be Ok
- provided the people who made them had permission

Stuff you create yourself will definitely /not/ be Ok

eg: you can stick up pictures, but you can't paint 'em.

Legally, you must get permission to use any trademarked character. If
you are selling licensed products, the manufacturer of that product
got permission and paid a fee to sell that character, so you can
re-sell it, but you can't make your own or use the character to
promote your site.

Just because you are not selling the Pink Panther does not mean you
can use him like a mascot, logo or icon.

I needed to know these things for my funny t-shirt site, www.tasteetees.com

I recommend the book, "Getting Permission, How to License & Clear
Copyrighted Materials Online & Off" by Richard Stim