According to this post, there had been people accusing them of using Morning Musume.'s name for attention, and they wanted to be recognised as their own group, not just a cover group. Musumen. had also always been a 10-member group, so when Zeararu. graduated and just 9 members remained, they changed their name and started afresh, keeping Musumen. a 10-member group forever.
Aoi and Nozakibento are the only members who have revealed their ages, and Aoi is the only member to reveal his real name. The other members sometimes joke about their ages, but all of them except Aoi were already adults when Musumen. formed, so don't believe everything you hear. If you see somebody claiming to know the members' names and ages, take it with a pinch of salt, because this information has never been officially released.
They aren't a BL group. MeseMoa.'s songs cover lots of different concepts, some of which can be interpreted as BL, but they have never claimed to be a BL group. The members have also never spoken openly about their sexualities except for in a Chinese interview back in 2014, when they said 'We don't know' when asked if any of them are gay. Like many idol groups, MeseMoa. have a love ban, meaning they're not allowed to have romantic relationships while active as idols. However, according to the official Twitter, the ban doesn't apply to relationships between the members. This rule was set in 2013 so it's uncertain whether it was a joke or whether it even still applies.
Not anymore. In 2016 the Sharehouse Squad moved into separate apartments in the same building, along with Nichan and Nozakibento; as of 2022, Shirofuku is the only one still living there. All the members live separately in Tokyo, except for Nokkuso who still lives in Osaka and commutes.
English is a compulsory subject at Japanese schools so they all understand it to an extent. All of them are more than happy to try speaking in English with fans at meet and greets, but for a proper conversation your best bet would definitely be Nichan, since he used to be an English teacher. From my experience, Nokkuso, Forgeru and Punchan are also pretty competent at English. Shirofuku has been studying English recently and understands a lot, but he's a bit shy when speaking it. Tomitake on the other hand is always up for speaking English, but his actual comprehension is...questionable.
Currently only the songs from the Sesshoseki Session, Karamete and May Say More singles are available here. Other groups from DD have slowly been adding their songs to sites like Spotify, so the hope is that MeseMoa. will start making their music available too. You can also find some of their songs at JOYSOUND karaoke venues.