1)   What is the dynamic here:  Does Saruman really wish to engage Gandalf and Theoden?  Would he have come out of his own will, or was he forced out by the command of Gandalf?

I think that Saruman came out of his own will. His last chance was to persuade either Gandalf or Théoden  that he still deserved a position of power. Not much hope for him in that, but it was all he had.

2)  A few of you have briefly addressed this issue in your previous answers, but I’ll ask it so others can elaborate: Why does Gandalf ask Theoden to join him?  What does Theoden hope to achieve by confronting Saruman?

I think that Théoden hopes for some sort of closure. The fact that he prevails in an encounter with Sarumant who took over his mind, caused the death of his son and laid waste to his country helps him maintain the self-confidence that Gandalf helped him restore.

3)  Realistically, what does Saruman think he can achieve by speaking with the King of Rohan?  Is he just trying to saving face in a bad situation?

Maybe Saurman thinks that if he could gain the support of Théoden and the Rohirrim,  he could rebuild his own army and start over. At least, any division he could cause between Théoden and Gandalf would interfere with Gandalf's plans, and interfering with Gandalf would give Saruman a measure of revenge.

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Although now long estranged, Man is not wholly lost nor wholly changed.
Dis-graced he may be, yet is not de-throned,
and keeps the rags of lordship once he owned:
Man, Sub-creator, the refracted Light
through whom is splintered from a single White
to many hues, and endlessly combined
in living shpes that move from mind to mind.
Though all the crannies of the world we filled
with Elves and Goblins, though we dared to build
Gods and their houses out of dark and light,
and sowed the seed of dragons--'twas our right
(used or misused). That right has not decayed:
We make still by the law in which we're made!