Is it a red flag to raise your budget by 2x, say from you got from $100 to $200 right after ad approval?
Do ads go into review queue again when you raise/change budget?
Ads go into review when you change any ad/ad set parameters other than bid (and maybe bidding mode, to be honest I haven't tried switching from CPC to oCPM but I presume it goes back into a queue) and budget.
So no, budget changes do not set your ads to pending. This would be the most annoying thing ever on a self-serve platform and it would disrupt any attempts at scaling, which would likely cost Facebook hundreds of millions of ad revenue per quarter.
As for red flags... raising budgets isn't going to do much unless you have a fresh account and FB has automated filters to detect people who create accounts in bulk. How these filters work is not something we know, but you can be sure an account that's 24 hours old that's raising budgets promptly is probably going to get a closer examination.
My account is 4 years old and I'm just started using it again for paid ads after 3 years.