...do (previous and/or multiple) rejections or time passed since offered affect anticipated acceptance in any way, shape or form?

and why does minimum bonus (requested by same player) change from offer to offer (+ or -) ?

If you cut a player, he will not sign with you no matter what for, what I have read, 2 seasons. The AI player "needs time" to cool off and stop being "IRATE" at you.
The minimum bonus raises during the first two seasons the more times you bother him. As the time lapse passes between offers, the highest current offer is the one that prorates down.
For example, the cut player first will not sign for a bonus of $100. Three games later he may sign for $70, but because you bothered him within his pouting time, his will reject the offer and make the bonus higher.
Its the Simulated World's little man's way of "Take this job and stick it. You need me more than I need you"

Yeah thanks, I kinda assumed (and have experienced) that so let me clarify my question...

When you're renegotiating contracts (with players who are currently under contract with your team and you're trying sign them to an extension)......do (previous and/or multiple) rejections or time passed since offered affect anticipated acceptance in any way, shape or form?

and why does minimum bonus (requested by same player) change from offer to offer (+ or -) ?

If you are talking about Gibson in USFL, I would say he wants Baker money

Gibson is a good example. He has rejected 2 attempts to re-sign. On my 3rd attempt, his Min Bonus "required" actually went DOWN. So, my question is is this a common occurrence? How often do bonus requests go down, and what affects this variable?

I would say that the season is halfway over and he is happy, so giving a hometown discount to stay on a winning team.
But of course, any advise I give you has a underlining intention of keeping the trophy at home ;)