I think Doraemon （a relatively old Japanese anime) is nice, not so difficult but they don't say things over and over. And all episodes are really funny.http://video.google.com/videoplay?docid=-780882...
I have ever watched "nihonjin no shiranai nihongo" (=the Japanese language Japanese people don't know), which I find interesting for all Japanese language learners.http://www.youtube.com/watch?v=iQn23l8N-yo
Oh, sorry, I forgot to answer your questions.
Question1. In the long term, if all my listening practice is from anime and dramas, is the input i'm getting going to be broad enough?
As for me, when watching TV dramas, movies or others, I review the grammar points or the new words I already learned in textbooks. If I encounter the words in different contexts, I will learn how to use them. The more I see them, the more active they will be.
Or do I need to try other things, like news, or variety programs, or something else?
Yes, if you find your favorite ones in different kinds of programs, you will be able to broaden your vocabulary