Hello

Good to hear the puppy is settling in.

Right, now for the issues.

Pups chew. It's what they do. They all do it and they all do it a LOT. The main thing is to make sure she can't get to anything dangerous like cables etc...and try to keep things you'd rather her not chew out of her way. Provide plenty of chew toys for her to chew on instead.
When you're not around, a crate might help to keep her off things. Her chewing will subside as she matures.
Telling her off can can cause her to chew on purpose in order to get a reaction out of you, especially if she wants attention. If she starts chewing furniture, introducing bitter apple is an idea but you have to make sure you do it properly otherwise it might not work out.
With hands, toes and all that, yes, all pups do that and all the pups we had just grew out of it without us really having to do anything. If you watch littermates playing and a pup bites another too hard, it will do one of two things.
The first is to start crying and try to escape and then refuse to play further. The other, especially if an attempt to escape fails is to lose their temper and turn on the other pup and give it a right going at. It was quite funny seeing a fluffy little pup muster up so much aggression on those occasions, haha. The one who was biting it would sort of back off and stop biting then the one who'd been bitten and had a go at it would turn and walk off in a huff. The pups soon learnt that biting too hard meant the other pup would be unwilling to play with them or may even go for them. If they bit an adult dog too hard, they'd also get snapped at. Because of these teachings, pups learn bite inhibition so that, when they are playing, they don't really bite each other at all or, if they do, they don't apply pressure with their teeth. When playing with humans, adult dogs don't tend to use their mouths at all, well ours don't. Even if you stick your hand in their mouth, they spit it out.

As for training, start now. Keep lessons short though. Maybe two or three five minute lessons each day. It's always best to start as you mean to go on. You can always get extra advice if you need it about the best way to get her to perform these commands.

As for her relationship with the other dog, it is common for older dogs to ignore the new pup or bully it a little maybe. Just leave them and things will sort themselves out. As the pup gets older, your other dog will develop a better relationship. Just maintain a pecking order between those two though and treat your older dog as the higher ranking of the two. The dogs need to know their place and it keeps thing peaceful and the dogs happy.
There's no point in trying to involve yourself in making them closer. This often causes more problems than it solves. If there is no aggression, I doubt you need to worry. Your older dog will come round in her own time. I've seen this alot myself and the older dogs always came round. In fact, it wasn't uncommon for those least tolerant of the pup to become closest to it when it grew older.