For a more modern look, yes, cover diagonal wood with drywall. Consider leaving some of beams that have nice character. Please remove the wood above the fireplace in the second pic. It is awkward and too large.

It is personal preference. But I wouldn't. I love wood and that looks like nicely done work. I see a fancy crystal chandelier and it makes me wonder if you are going to change that or you're going to go rustic....or want to modernize or transition to something else....I think the stone of the fireplace is not working with the wood. The wood is more modern and the stone, really rustic. Maybe this is what is bothering you?