The word 推し (oshi) in the context of idol culture comes from 推す (osu) which means "to push/to support". The idea is that when someone is your oshi you push for/support them to succeed and reach the top, etc.
Aria's line here uses the phrase 私が推すもの (watashi ga osu mono). A more stiff...
I don't believe they've explicitly mentioned anything about the popularity vote being rigged by Tsubasa (or anyone else for that matter). So far the current Top 3 lineup seems to be organic
We took a hiatus at the end of last year and have been slowly restructuring while catching up on the chapters that came out. But we're still working on it, and will be releasing a chapter relatively soon