"Should we do it?" is acceptable in AmE. 'Shall' is practically non-existent in AmE outside of legalese or overtly formal language. I can't speak to Canadian English, but I wouldn't be surprised if it's similar to AmE in this aspect, as it is in so many other cases.
While I can't say who generated the subtitles, I can say they weren't auto-generated by YouTube. There's no option to turn them on/off, so they were built in to the original video before it was uploaded.
2- "Should we do it?" is wrong there. I suspect he wanted to use "shall" instead of "should". I think he was intending "Let's do it".
Note that "should we do it" is possible, but it is a genuine question rather than a suggestion/invitation to do something.
While that may be true in BrE, as Tarheel and I have pointed out, 'Should' works in AmE (and by extension, presumably Canadian English). In this context, 'should' does indeed carry the meaning of the blued words in AmE.