Thanks for your message! I'm glad you're enjoying the game. Laser Squad was one of the main inspirations, alongside the Traveller TTRPG. At some point I might to a "rules" document that details everything as if the game were a tabletop RPG; I was thinking of it in those terms when I designed it. As for your queries:
(1) Two dice are rolled for a skill check, one must be equal to or under the skill level, and 1 on either die is a critical success if the skill in question is level 6 (since a double 6 counts as failure for even level 6 characters, the critical success mechanic is what separates level 5 and 6 skills).
(2) There's a check for each step when you're moving. At the end of your turn, there's an extra stealth check to see if your character stays "alert" during the next turn, which allows them to fire immediately if an enemy appears in their line of sight. Normally a character has a 1-turn reaction delay during the opponent's turn.
(3) Creeping status ends when a character's side's turn ends, after which they're either "ready" or "alert". So nobody's creeping when the enemy does their detect action.
Training will definitely improve things, especially as you find interesting equipment that nobody's particularly competent in. My usual strategy is to recruit cheap characters at the start to maximise money for equipment, then gradually train them up.
There's also item study in the equipment screen, which will be useful for making use of alien items if you recruit anybody with the Xenotech skill.