1. I have not seen this. The skip could be caused by something gone wrong, some sort of error that's causing the time period to skip (like the issue earlier that caused all characters to get sick). If you know how to, look at the error log for errors when it happens. Maybe there's some indication there of something going wrong.
Where's the error log? Should I run the game in dev mode to generate it?
2. Maybe she worked out a lot in the in-between. Endurance can be raised that way although 25 points seems unlikely. There's a couple of events that raise someone's endurance to fit some story need. In any event, endurance only counts for 20% of the janitor job skill which actually translates to around 3-4% of the job skill. Experience (how long they've been working a job) counts for about 30-40 points of the job skill value. So I don't see anything odd about those values with the info you provided. And your bad cook having an effect on satisfaction that also depends on the price of your food and how early is in the game. Bad workers have a smaller effect early on than later in the game. Job skill also affects how many customers can be served which again matters less early on.
No, it was a matter of weeks. Clearly a case of AI cheating

I didn't know experience & education were a factor in calculating job skills, and it makes perfect sense. And I should have hovered over "Job Skill weights" to find out... btw, what does "attract" means for jobs like singer, stripper, etc?
4. It can happen due to actions, like you get their wife or gf pregnant, he was trying to get together with someone who wouldn't because she was involved with you, you had sex with their gf or wife, you got into a fight with them (not sure this can happen with the MC, but applies to other characters), if they discover you tried to mess with them (like sabotaging their club), or if their opinion of is negative they can decide to do it (the odds are higher the worse their opinion of you is as well as their compassion).
I'm positive all my rivals/enemies start with having a "different ethnicity" negative modifier, possibly followed by a couple of negative idle chats. One NPC (black, female) in particular seemed to have a lot of rival/enemies who were white and male.
If you don't want to turn it into "Race Wars", I suggest you tone it down a bit
All of those are good suggestions. Managing 3-way relationships are very complicated so that's why a lot of those things haven't been done. The last part is really good idea, other than the specifics of it, the bar previous owner was your father and the bar had been closed for a number of years. But yeah, incorporating some sort of story like that into the tutorial, I like it!
Once again, imagine this:
First scene: you and the other club owners meet with the Commissioners; they explain that, "when she has time", the city inspector will visit your club and see if it complies with all the rules & regulations etc. ( = you meet a lot of major characters from the get-go instead of having to look for them; you introduce the inspector from the get-go: the loan forgiveness law is ofter passed before the inspector is created, which makes little sense; you keep some randomness because you don't know when the inspector will visit you, you can be the first or the last in the line)
Second scene: you call your sister/daughter and ask if she can help you setting up the club, she agrees and promises to work as a bartender until you'll find a better candidate (You find something to do with the sister/daughter who is "just there", at least for the first few months; she can act as the "civil servant for the club", basically giving info for unexperienced players)
Third scene: The inspector eventually visits your club and gives you the green light (maybe after you bribed/seduced her...)
Fourth scene: after some weeks, you have to decide whether or not you want to hire your sister/daughter permanently. Of course she may accept or refuse your offer.
Also, have you considered a proper character creation? Full date of birth (instead of age) + an "archetype" + starting image. For instance, a 20-something fit man who is a "Playboy" would start with good physical stats, very good sex stats, decent social stats, bad mental and work stats; a chubby "spoiled brat" would start with more money but overall bad stats, and so on. And you can use the day/month of birth for further randomization (you know, the zodiac, stars alignment...)