Agreed. the season was underwhelming but if you ask me, only the first one appealed to me and that too mostly because of the novelty. Anyway, here are the answers to your questions
1) How did Eurus go back in jail?
She practically controlled the facility and came in and out as she wished. Everyone over there was under her control.
2) How did she even get out?
see 1. she put the governor under trance. used the speaker system to control the all the people in the facility.
3) How did she get those 3 brothers and the Governor's wife
through her resources which were obtained by mind controlling powerful people like the governor. you have to understand that every person in that high security prison was under her direct control. flying in and out and getting a few people wasnt an issue at all
4) How was she a genius and how do you even "reprogram" someone you meet?
she was a genius just like her other two siblings. technically you can reprogram a person through hypnosis or other mind altering techniques like resonating special frequencies that activate certain enzymes etc. in the body to make human beings more susceptible to control. how she did it was never specified, maybe some advanced form of hypnosis that she was born with. call it her superpower if you feel more comfortable with the term
5) How did she end up in the cabin in the end where she was controlling the damn prison!!
i presume you are talking about the end. she made then unconscious and took them to the cabin possibly with the help of staff at hand who were working for her.
6) Watson's feet were chained to the bottom of the well - how did he unlock it when thrown a rope?
that part wasnt shown. only him holding the rope was shown. possibly, a tiny detail left ambiguous since there was help available anyway
7) Why did Eurus pretend to be Faith and John's therapist - what was her end goal?
boredom. she did things just because she wanted to. no particular reason at all. also she predicted all the things as they'd happen and continued to set sherlock puzzles that he kept unlocking
8) Why were Lestrade and Molly so underutilized as compared to the first few seasons?
because the story line has become bigger and doesn't fall in the same premise anymore. big is better and fancier not just solving ordinary murder trials anymore.
Going back to season 3
1) How did Moriarty play that "Miss Me?" video all over the country!!! What the hell was the purpose of it?
the end game was eurus playing around with the heads of her brothers.
2) Did he even die in the first place?
yes. he died. he always wanted to die a heroic death and become part of a major plan to continue his game after death - an opportunity that eurus provided him perfectly.
3) What was Moriarty's grand master plan in the first place (Lady Blackwood asked about it in the first episode but freaking Sherlock decided tweeting was more important...
the grand master plan was boredom. an extremely intelligent being stuck in a mind prison with no way out. and ultimately sherlock helped her escape the mind prison.
the show wasn't good and ever since the novelty factor has gone, the holes appear bigger. good or bad, as dr. watson said, it is what it is.
