First of all, you have to make sure that your father (or his father)
was indeed a German citizen, and not just born in Germany. Unlike the
US law (where anyone born in the US is basically a US citizen), German
citizenship is inherited, and there were many people at the turn of
the century, who lived in Germany but were in fact workers or refugees
from other parts of Europe.
If your father was a German citizen, you are also entitled to be one,
and only have to apply. However, to see if that is the case, also
German Citizenship (PDF)
Assuming that you're through this first stage and you've established
that you're eligible, as for your questions:
(1) "Both the United States and Germany recognize the concept of
multiple nationality." [...] ". In the past (I'm talking about the
beginning of the century) the United States demanded sole citizenship
and frawned upon dual-citizenships. This is, however, over - also
through court decisions, and many Americans hold dual citizenship.
Please note, that if you get in trouble in Germany, you cannot
disguise behind the American consulate or even seek their assistance -
for the Germans, you are a German citizen.
Dual Citizenship FAQ
Dual Nationality (State Department)
(2) There is no such thing as a "EU Citizenship": the EU is a union of
sovereign counties. The citizenship, however, is a citizenship of an
EU member. That means:
- You may work and live anywhere in the EU
- You may apply for vacancies in the EU open only for EU citizens
However, you can?t, for example, vote in the national elections in the
UK (even if you?d live there), because you?re not a British subject.
I hope this answers your question. Please contact me if you need any
clarification on this answer before you rate it. You may also want to
read my previous answers in the subject:
Downside of Dual US/German Citizenship?
Rights to Live in Work in the EU for a US citizen....
Obtaining German Nationality through marriage