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?
Saruman very probably wants to have another go at Theoden. I don't think he's
quite realized yet the extent to which Theoden has escaped his clutches. As
others have pointed out below, Theoden, having been under Saruman's influence
for so long, is vulnerable to fall under it again, and Saruman knows that and
wants to take advantage of it. If he could have had him so strongly under his
sway from a distance he probably thinks it might be easier to make Theoden
succumb when he's right there in front of him. In my opinion, however, I think
that, having been burned once already, Theoden is one of the least susceptible
to Saruman, so Saruman is making a tactical error right there in hoping to
seduce Theoden again.
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?
Remember that in the march to Helm's Deep it's evident (or is it? we never
really agreed on the other soldier's reasoning) that Gandalf still does not
have the support of all the Rohirrim. Saruman's voice was described as
something that, when not directed at you, is easy to see through, and you smile
that others cannot resist it. Having that voice directed at Theoden but not at
the rest of the Rohirrim who did not go up to Orthanc will let the men have a
chance of seeing Saruman at his worst to reduce any chance that they might
still consider him (and Grima) an ally. Also, Saruman would be a valuable ally
if they could sway him, so they need to try for their own sakes, and for his
sake. Repentence is a big controlling factor in Middle Earth: think of Boromir,
whose repentence would alone have justified having brought Merry and Pippin on
the quest, even if they had served no further purpose, or Gollum, whose
repentence is in Tolkien's own opinion one of the most important moments in the
story. You can't be a good guy in Middle Earth if you don't try to save your
enemy.
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?
As I mentioned earlier, I don't think Saruman realizes yet how free from his
influence yet Theoden is, and still hopes to regain Theoden's mind. Gandalf can
do nothing to stop him. Gandalf is a steward, a protector, and cannot forcibly
make any of those he must protect do anything against their will, even if it is
for their own good. If Saruman can get Theoden, Gandalf's being there will do
nothing to prevent Theoden's fall, and it is not likely that Saruman will get
another
chance.
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Then Fëanor laughed as one fey, and he cried: 'None and none! What I have left behind I count now no loss; needless baggage on the road it has proved. Let those that cursed my name, curse me still, and whine their way back to the cages of the Valar! Let the ships burn!'
