Regarding Gríma and Éowyn

[Bombadil]

Are you kidding? Eowyn is Bad @$$! I have full confidence she'd kick Grima back to Isengard. Let's not forget she killed (half-killed) the witchking!

[Evisse the Blue]

In that scene from TTT with Eowyn and Grima near Theodred's death bed, when he 'sweet talks' to her (using some of Aragorn's words from ROTK, as it has been already remarked on this forum), don't you get the impression that she is almost about to give in? But then she suddenly recovers, as if breaking a spell. Grima surely had a way with words, and Eowyn was, although very courageous and noble, sad and lonely. So, I think MovieEowyn might, in an alternate universe, given in to Grima, in a moment of weakness, and be very angsty about it afterwards.
But, without her consent, I don't think he would have been capable of overpowering her. Even if he is stronger than her (which I doubt), she knows how to fight better than he does.

3. In his own strange way, he really does love her, and he would not take her unwilling-- he really wants her to love him all on her own.

I don't think he ever harboured such hopes (if he did, his sense of reality is really low), nor do I think that he loved her for who she was. What he felt for her was simply lust (like Morgoth felt for Luthien). The baddies is Tolkien's world don't seem to be capable of feeling love.

[Eomer of the Rohirrim]

Remember that Grima was a Man of Rohan, not a cartoon gnome like movie-Grima. Anyway, Grima was extremely sly. He apparently poisoned Theoden remember? I'm sure he would resort to such devilry if he wanted to do the deed with Eowyn.

I don't think he'd ever pluck up the courage to do it though.

[Imladris]

As I recall, in both book and movie, Eowyn was the promised prize to Grima when Saruman killed Rohan. Therefore, maybe he was not expecting to overpower her himself but that grief would do it for him.

As Evisse pointed out, she was almost on the point of giving in (at least that's the impression I get) and that was when she was in grief.

Cheers,
Imladris

[Encaitare]

You think it was just lust? You could be right, but I, ever the hopeless romantic, think it was a bit more than that, even if Grima's sense of love is extremely twisted. He actually understands her, more so than anyone else and I personally think he loves her spirit as much as her body.

[Lalaith]

I found the 'almost giving in' scene in the film extremely distasteful and I think Tolkien would have hated it.
As for being offered to Grima as a prize when Rohan had been destroyed, BookEowyn would, I feel fairly confident, have fallen on her own sword before being dishonoured in such a way. And may well, of course, have finished Grima off while she was at it.

[The Only Real Estel]

Are you kidding? Eowyn is Bad @$$! I have full confidence she'd kick Grima back to Isengard. Let's not forget she killed (half-killed) the witchking!

But she didn't beat the Witch-King in an arm-wrestling tournament. No doubt Grima could've killed the Witch-King if he fell right in front of him (if not for the prophecy).
If Eowyn had ready access to a sword, no doubt she'd have Grima's head on the floor quicker than you can say 'nice try', but if she had no sword it could well be a different story. I'll agree that she is 'Bad @$$', I'm not taking that away from her, I just find it hard to believe that she could've bested Grima through brute strength. Again, I'm not trying to take anything away from Eowyn.

I found the 'almost giving in' scene in the film extremely distasteful and I think Tolkien would have hated it

Same here. Eowyn (as Tolkien intended her) would never have even almost given in to Grima no matter how much she 'thought he understood her', & would've seen through his petty masks. Mostly thanks to what she'd already seen him do to Theoden.

BookEowyn would, I feel fairly confident, have fallen on her own sword before being dishonoured in such a way. And may well, of course, have finished Grima off while she was at it.

No doubt about it!