Honestly, it sounds a bit like you might be going a bit too fast for him. Easy to do when they are adult dogs because we have different expectations of them than of puppies.
Take the crate training. If you have only had him a few days and he has never been trained before and kept outside then asking him to be in the crate is far too soon. If he would dash in and out just to grab a treat from in there, that would be great progress in my eyes.
Does he sleep if you are by his side? If so, can he sleep in your room for a few nights while he settles or can you sleep downstairs with him for a bit? These are the kinds of things you do with puppies in their first few nights and I would suggest you treat him exactly as if he were an 8 weeks old pup. Lots of soothing stroking and soft words, like with a baby. Even better if you can do all this with him where you want him to sleep eventually.
Clinging to new owner's side is a fairly typical behaviour for rescue dogs, at least at first. He will need work to slowly build up his confidence away from you. Baby gates can help because he can see/hear/smell you but be contained in a neighbouring room. Give him lots of treats and toys in his room so he can slowly learn to enjoy himself away from you. Do lots of leaving him in a room with a treat for a few seconds on his own before returning - all builds up experience of him being ok without you.
In all honestly, neutering sounds like a gamble. It has a real risk of making the behaviours you have described worse:
- being taken away from you for a worrying experience may make him even more clingy when he gets back
- having his testosterone removed may leave him much more nervous
- it may stop the marking but probably won't, neutered males mark and by now it may be habit with him, which means it is not chemically driven and so neutering won't make a difference
It's tough because you have all the challenge of a puppy, but in a fully grown dog. He's going to need patience, perseverance and kindness to get where you want him to be.