[Adam]

Again, I highly recommend getting a rule book. Rule 4 is where you should start, Definitions. That said, LGP is gained when a defender has two feet on the floor facing his opponent. Note, there's no time requirement, nor is there a requirement that he be stationary to gain LGP. Once LGP is gained, it may be maintained by moving sideways or backwards as long as the player is not moving into the offensive player when contact occurs.

I wouldn't call it an "illegal screen." Call it a block or a push or (maybe) a hold.

Defending a player with the ball is different than a player without the ball. On ball defenders have more rights than off ball, as far as movement is concerned. Off ball, screening principals apply to both sides. On ball, the defender does not need to provide time or distance as long as he is either in LGP or stationary.

Blind screens off ball must provide time and distance (no more than 1 step) or the screener can be called for a foul. If a blind screen has provided sufficient distance, then the contact may be severe without a foul so long as the screened player attempts to stop upon contact.

You are correct, running through a screen is a charging foul.

There is no "illegal screen" mechanic. It's a foul, either blocking, pushing, charging, or holding. Is the mechanic that fits the nature of the foul. If you verbalize it when you report, don't say "illegal screen."