1) Don't really think so. As in hor, it will be (non - tmall i.e official stores) selling the battery. They can claim its original but you won't really know cause it's just a sticker.
(If u die die want official original battery best is go back Clark quay and repair , I think with diagnosis fee etc would be around 50 SGD) my friend just changed the mi Max battery from them not that expensive , around that price iirc.
Xiaomi taobao official store don't sell parts for their phone.
2) regarding side buldge I really can't tell you what is the problem / cause. You need to personally open it up to take a look. (If it's because of the battery the only reason I can think of is that it has caused the metal back to deform due to the buldge and because of that creates a gap at the side. (Btw you can also buy a new back cover with buttons for I think less than 20 SGD from taobao, just make sure correct model as got people say the Kate kenzo got different length etc. (Super confusing , I know)
