caulk wll shrink and not hide the holes, just makes them look "different"
You can use a wood filler or a spackle.
When painting the wood, we like to use Minwax wood filler in the toothpaste tube. Never use caulk and the putty will also shrink.
Great picture you have!
Yes, either use the wood putty or shrink-free spackle to fill the holes.
I like to use hot mud, it dosen't shrink & dries quickly