If you have a dog club close by check out and see if they run obedience classes. They are usually inexpensive and a good way to both socialise your dog in controlled company and learn some obedience training and they are usually fun too. Labs are lovely intelligent dogs. We had quite a few naughty lab pups come through the obedience classes our dog club ran and it was amazing the difference it made.
I dont tolerate biting, as I have had a couple of mouthy cattle dogs. With one I taught her she could be mouthy if I had a pair of thick gloves on but no way on human flesh. I just said ouch in a very dissaproving tone and immediately stopped all playing. My dogs soon go the idea that all fun stops immediately if they engage hard on my flesh. Deflecting them on to a tug is okay but if they dont learn that flesh is off limits they will often latch on to the hand holding the tug.
I found crate training puppies is a great way to give them a place where they are contained and can learn to chill out.
Once she starts to learn some manners and boundaries and you introduce some structured training, the barking should ease up. Engaging their brains will often tire them out. I just dont engage with my dogs if they whine and bark and they soon learn that such behaviour gets them nowhere fast. I am very quick to reward and praise good behaviour even if I havent asked for it. If they are lying quietly in their crates I tell them what good dogs they are.
Dogs are often good with routine, and will learn to chill when they recognise that it is not their time for walks, games, training etc.
Do not let her drag the cat by its neck. I would actually be pretty firm with that, voice your dissaproval when you recognise the impending signs of roughness before it starts and stop her. One of our cats was killed by one of our young dogs in that manner, and there was not a mark on her, it can happen very quickly and instantly and without it looking particularly rough and a lab is a lot stronger and heavier than a cat and a neck can dislocate easily enough.
It is a very dangerous activity for your cat even if the cats appears to enjoy engagement with the dog, our cat did and I am sure the dog didnt mean to kill her. They got on well together. We learnt the hard way.