I don't think we know why Saul chose the particular occupation he did, so I'm not sure that Questions 1 & 4 are answerable.1 We don't know all the details of Ed's extraction process; maybe the occupation was "baked in" (hah!) to the new identity that Saul was given, and he just went with the flow.
We have a bit more to go on concerning Questions 2 & 3, though.
In "Breaking Bad" (S6E11), Francesca tells Saul that the feds have seized all of his assets, "except what you took with you — and I'm guessing that's not chump change." We actually see the cash that Saul takes with him in "Quite A Ride" (S4E02),: enough to fit in a small bag, about 8" x 8" x 12" at most, and not particularly tightly packed (a bunch of rolls of cash.) If I had to guess, I'd ballpark it to be on the order of $100k, give or take a factor of two — substantial but not enough to live off of indefinitely. He also took some diamonds with him, but those would have been difficult to liquidate without drawing attention. And even if he had a lot of money, living a lavish lifestyle while having no visible source of income is a good way to draw the attention of the IRS — exactly what he doesn't want at this point.
As far as Question 3 goes: Gene does start scamming people instead of relying on his Cinnabon wages. Gene only actually works for Cinnabon for about six months; per this timeline, he "disappears" from Albuquerque in March 2010, and he's running an identity-theft scam by mid-November. But how does this turn out for him?
In the end, Jimmy/Saul/Gene is too undisciplined to fly under the radar for long when doing criminal activities. Within about a month, he insists on going ahead with a scam when it's obviously no longer viable, leading directly to Jeff's arrest, which in turn arouses Marion's suspicions and leads her find Saul's old ads online.
1 At least in-universe. Out-of-universe, Vince Gilligan & Peter Gould probably thought it'd be funny to have Saul's throwaway line near the end of Breaking Bad come true:
I'm not your lawyer anymore, I'm nobody's lawyer. The fun's over! From here on out I'm Mr. Low Profile, just another douchebag with a job and three pairs of Dockers. If I'm lucky, a month from now — best case scenario — I'm managing a Cinnabon in Omaha.