for the best newspaper, which tries to stay neutral and cover facts in deep, I would go with "die Zeit" - a weekly paper. Frankfurter Rundschau and Frankfurter Allgemeine are good daily papers, the first more left leaning, the second more conservative. The FAZ feuilleton has the best reputation in cultural articles.
The Süddeutsche used to be good, but went down in quality. Same with "der Spiegel" - but occasionally you find some good writing in both.
The good thing about Der Spiegel is that they have all articles in their archives, it is pretty good for research purposes of past events.
Deutsche Welle is good - and they pay well
They also try to be unbiased, as an alternative you could check out tagesthemen, a daily 30 minute tv news show, which tries to focus on the topics of the day, their website is ok, you can find much coverage there.
Personally I like the "Jungle World"
- which is an undogmatic leftist weekly paper. All articles are online after a week.
regarding German news on English, you might want to check out thelocal.de, a site for english-speaking immigrants, they have quite short coverage, but may be a good start to see what's up in Deutschland.