With your A&P certificate you can sign off anything from from a Beechcraft to a Boeing! You don't need a type rating. If you're on the airliners, your company will dictate what you can and cannot sign. You are effectively an extension of Maintenance control, you cannot deviate from their instructions.
There is no avionics licence per-se, but there are specialists that would fall under an FAR145 establishment.
Your IA lasts for two years and has to be renewed every odd year in March. Next renewal for all IA's is March 2011. In order to keep your IA valid you must have performed at least 4 annuals, or 4 major repairs or alterations or a progressive inspection on an N reg in a 12 month period. If not you will have to attend an FAA sponsored IA renewal course.
EASA does not exempt any of the written exams for B1/B2 with your A&P, but they might take the experience into consideration.