Regarding Gríma and Éowyn

[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!