Yes, goats can get into things if you don't put them up or fence them off. The goats I have been around weren't that bad but, I never really was around a male so can't speak to the trouble they are. The nannie goats with kidds were always lots of fun and the ones I was around where a bit like pet dogs that would come when you called or like my Polled Hereford cows, would come to the sound of feed in a plastic bucket "Pied Piper" style. Left to their own in substandard fencing, they would 'escape' but, being feed trained made them easy to round up when the fence was fixed.
Regarding browsing shrubs and brambles, sure they won't graze the woody parts to the ground but with no leaves they aren't going to do well in most soil types.
While I wouldn't mix livestock types with them, they were generally on common pasture with my cattle and I never noticed any undue health issues from that.
Perhaps the goats I was around are different and heartier than those commonly found in the UK? For me, I wouldn't call them delicate at all but, like cattle, it is likely breed/genetic differences that we are comparing. A Longhorn, Brahma, Hereford, or ... are all pretty drastically different in where they thrive, how they grow, and what their temperament is. This doesn't even touch dairy breeds.
Sheep are the most problematic for me but, I don't live in sheep country so, my experience isn't really valid and is very limited unless we are talking about one on my plate purchased from the store.