Are there any known lists of monthly/annual readership (e.g. top charts) of different mangas in Japan or the US? Perhaps by category and age groups (audience)?
What are some good English sources for looking up statistics of manga readership and reviews?
In Japan, I don't know of any chart which tracks readership, and statistics for that seem hard to gather. The next best thing is sales numbers, which represent how many volumes were bought, but volumes purchased do not necessarily directly correspond to readership. Comic sales numbers are reported on a weekly basis by Oricon, which is the go-to source for Japanese entertainment media rankings. These are reposted across the internet, including by Anime News Network in English (link to current most recent list). As one would expect, the weekly rankings tend to vary a lot depending on what has been recently released, so they also compile half-year sales statistics (English via ANN) which are probably more useful for determining what's currently popular.
Overseas, even sales statistics aren't going to do a very good job indicating what's popular because of the levels of manga piracy. They are still available for the US via NY Times, but they miss a significant number of viewers who are reading pirated manga (e.g. scanalations) and also don't report absolute numbers, only relative positions. Various indexing sites like ANN or MAL do have data for "most popular" manga based on their listings, which likely includes those who are obtaining it illegally, but these are all-time rankings. Current rankings are probably difficult to compile with the data they have.
