May I fly an aircraft registered in an EASA member with an EASA licence (NOT same country as airplane) outside Europe?
Under ICAO rules, if Germany 'validates' your Italian license then yes, you can fly the German aircraft in other countries:

Use of flight crew licences on foreign-registered aircraft
Any pilot
who wishes to fly on an aircraft registered in a State other than the
one that has issued the licence, needs to obtain an authorization from
the State of Registry. This authorization is generally given by the
State of Registry through a validation or a conversion of the foreign
licence. In general, the validation process is used for short-term
authorization while the conversion process is used for longer-term
authorization.

What 'validation' means here is up to the country where the aircraft is registered. ICAO itself doesn't define what they should do:

When a State validates a foreign licence, it recognizes it as valid
for use on aircraft on its own registry. The Convention on
International Civil Aviation and its Annex 1 do not contain specific
requirements for the validation of licences beyond establishing the
principle and the fact that the validity of a validation, cannot be
extended beyond the validity of the supporting licence.

In the US specifically, these ICAO rules are implemented in 14 CFR 61.3 (emphasis mine):

(b) Required pilot certificate for operating a foreign-registered
aircraft within the United States. No person may serve as a required
pilot flight crewmember of a civil aircraft of foreign registry within
the United States, unless—
(1) That person's pilot certificate or document issued under §61.29(e)
is in that person's physical possession or readily accessible in the
aircraft when exercising the privileges of that pilot certificate; and
(2) Has been issued in accordance with this part, or has been issued
or validated by the country in which the aircraft is registered.

Practically speaking, I would say that if you have some official piece of paper from the German authorities confirming that you are authorized to operate German aircraft - or at least this German aircraft - then you're good to go. The piece of paper may even be unnecessary as far as Germany is concerned, but if you plan on flying internationally then I think the more pieces of official paper that you have, the better.