Neutering probably won’t make any difference to trying to hump btw, it’s an over excitement behaviour not a sex one. Just so you know.
My dog was 5 months when I got him and had been in a shelter that entire time, so I can give you an idea of the things he found problematic, though obviously some might just be him or his age.
Stairs, he’d never seen them, couldn’t do them and was convinced they were evil, and he didn’t want to be left alone so I had to sleep downstairs until I could bribe him up all the stairs.
Things that make loud noises... washing machine, hoover, hair dryer etc. Those actually weren’t too bad, I think because I expected him to not know what they were so did them really gradually and made sure he knew it was me putting them on and off, not just that they make sudden noises. The hoover especially because so many dogs have an issue with it, I pretended to hoover with it turned off for about a week 😐😂 till he was not bothered by that, then spent time telling him I was turning it on and off... I mean, I’m absolutely not saying he understood me btw 🤣 just that because I was talking to him and very obviously OTT pressing the button he could see it was me making it do things not just some sudden thing. He’s totally unbothered bu all of those so it worked even if it sounds daft.
Tv, radio...on low and gradually, lots of pausing or turning things off because the noises scared him. The odd thing still freaks him out, men shouting, explosions, he’ll look for the barking dogs... but he’ll also happily watch dog tv for about half an hour, so it’s not all bad.
Reflective surfaces... took a long time to convince him they’re not other dogs.
Chewing, this might not be an issue for you as he’s older... but mine pretty much worked his way round the house systematically trying everything out for chewing potential, furniture, walls, the stone hearth 😐 just treated it like the big version of tiny puppy chewing and he’s pretty good now, though he has more stuff for him to chew on than most dogs, we have to be careful with his bedding and soft toys... and he will still steal paper to shred occasionally (he’s over 2 now)
The biggest issue we’ve had is that after coming home he developed a huge issue with other people coming in the house, that’s very probably to do with his background, but just to warn you that big things can suddenly crop up once they’re home.
I’d expect yours to possibly have an issue with food if it’s been a limited resource, so I’d be watching for stealing food, bins etc, but also from your other dog and resource guarding over food... obviously he might be fine, but they’d be what I’d be watching for.