To summarize for the OP:
Yes
Regardless of what you call it, at that point the DP is out of the game.
No.
Correct.
And to think this is one of the simpler scenarios
can a bench player sub in as an offense-only player for the original DP, which, would make the new player the new DP?
Yes
Scenario 1: Flex comes in to run for DP.
Question: To my knowledge, this is a substitution, meaning the DP is now out of the game.
Regardless of what you call it, at that point the DP is out of the game.
If I want to bring the DP back into the game on offense, can she come in for any player other than the flex in the batting lineup?
No.
Or can she re-enter only in the spot she previously vacated in the lineup?
Correct.
And to think this is one of the simpler scenarios