When is the earliest point in which i can start a new cycle

Well, as Artless indicated, Mystery Rooms show up every 3 floors.

Can you increase the probability that one shows up on every 3 floors? No. The chance is fixed.

You can, however, "fix" yourself a 100% probability if you have a spare Mr. Gency's Exit. Once you have cleared 3 floors (wherein the first floor had a Mystery Gate), then the next floor is eligible for a Mystery Gate if it isn't a boss floor. If you don't get a Mystery Gate when you first enter that floor, then use the Mr. Gency's Exit to leave the Item World, and save your game. Upon your return to the Item World, the floor will change, and a new chance for a Mystery Gate is rolled. If this also fails, then you reload your save and try again. It usually doesn't take that many tries, the chance isn't particularly low.

If you're trying to maximize an item by getting every Mystery Gate, you're going to need a lot of doors and a lot of patience. So make sure to save up!

Speed and Hit are the primary factors, though facing has a minor effect on accuracy (I've never noticed exactly what %), and some Evilities affect accuracy (such as the Ninja having 50% base dodge change if attacked from the front and Mothman having a 50% dodge change for projectiles or something). Weapon types have no affect on accuracy beyond the basic HIT bonus/detriment they may or may not have.

As for a specific formula I'm afraid I don't know. I just know that generally your HIT should be within the ballpark of the target's SPD or you'll either be sure to hit or sure to miss. Stealing has a much higher HIT requirement than simply landing a skill, too.

Also, the games will generally display "99%" when they mean "100%" accuracy. I'm fairly confident 100% accuracy is actually possible, as I've played enough Disgaea I really should have seen 99% miss, and I haven't.

Best I can tell the hit/dodge stuff has been mostly the same per-game, though later games have more Evilities that can affect dodge rate (and before Disgaea 3 the games weren't always transparent about what special attributes, now called Evilities, classes had).