I would like to know if with the Activity Activity API I can read more than the last 2000 tweets or read tweets from more than three months ago and if we can get it when an RT is done, a favorite is given or any type of event happens on a tweet and if you have the possibility to get filtered results.
Historical Tweets or ReTweets from a single @handle would be a better use case for the Search API using the:
from:handle OR retweets_of:handle
operators, see here for more details: https://developer.twitter.com/en/docs/tweets/search/overview/premium
Favorite/Like or other events for Tweets are only served through the Account Activity API.