I think there are a lot of good answers here, but I still wanted to add a couple of things. One, I think several of the posters are correct in saying that you should yelp or say "ouch!" loud and sharp and quit playing. That is how puppies socialize with one another. If one plays too hard, the other puppy lets them know this way and quits playing with them. They learn that they don't get what they want by doing this.
Another good way to help teach a puppy this is if you can find the right older dog. I've had two that were good at this. A puppy would try to play and they'd ignore them. When the puppy got too rambunctious they'd growl, if they didn't desist immediately, they'd snap and knock them down. They didn't hurt them or actually make contact with their teeth, but they shocked the puppy and made their point. Typically, the puppy would look shocked, hunker and come back licking at them, and if the dog growled again, they'd go off and play somewhere else. They didn't act afraid of that dog, but they definitely respected them more from that point on. That's what you're trying to imitate. You warn them, then deal with the issue in an appropriate way.
Another example are two of my horses. I've had them since they were born and they're full brothers. Both of them were with their mother in a pasture beside another mare. The babies both worried their mother to death, jumping on her, biting her, etc... She'd squeal and hump her butt up, but wouldn't every actually correct them. When they got old enough, the next door mare they grew up with would be the first one to get turned in with them. She was sweet and wasn't an aggressive mare, but she did protect her personal space. Both babies, a year apart, did the same thing. As soon as she was let in, she walked to a patch of grass and started grazing. They ran right up and reared up on her back and bit at her neck like they did with their mother and she let loose and kicked them in the chest. They all ran around for a while and settled down. The baby came back, being apologetic to the other mare and kept getting closer. She warned them several times, but when they touched her, she let loose again. They were stunned, but they listened to her signals from then on and also paid attention to other horses when they signaled this way. She wasn't trying to hurt them and they were more shocked than hurt, but they learned a lesson.
Now, I'm not telling you to hit your dog. The point I'm trying to make is that you have to find a correction that fixes the issue in a way the dog understands. I don't believe tying them or putting them in a kennel solves the issue. I think letting them out when they whine is creating another issue that you won't want.
Basically, try to think of how you want your perfect dog to be and work on it a little every day. To fix the biting issue, I'd do a couple of things. One is be consistent with whatever you do. For instance, if a dog begs for food and you say 'no' one time, but slip them a snack anther time, or someone else does, then you've basically taught them that they should keep begging, because sometimes it works. If you always consistently say 'no', then eventually they stop asking. Another thing to do is to make sure the dog is exercised, mentally and physically. Playing in the yard burns off physical energy, but you need to burn that mental energy as well. Structured walks, where they aren't allowed to stop and do stuff at random are good at this. They get physical exercise, but they also have to focus on matching your pace and watch for your turns. Puzzle toys can help as well. A tired dog isn't as liable to jump and bite. Granted, as a puppy, the stamina isn't going to be great, but we have a 1-mile walking path and when I have puppies I take them on it. When they seem tired and lagging, I carry them. When they squirm I let them walk. It doesn't take long before they can keep up the whole mile and are ready to out pace me.
Next is something hard to describe. When you correct your puppy and have something you don't want him to do, such as jumping and biting, many people say 'no', maybe they even yell/scream it. However, you have to wait for the dog to give in. A good example of this is a dog that focuses in on a cat or squirrel or whatever. The dog focuses in and tries to go for it. The owner tugs and pulls on the leash. Then they get in front of their dog to body block it. Then they walk toward it and back it up. They say they stopped their dog and they're doing some of the right things, but the whole time, the dog is focused beyond them on whatever it is they want and they haven't given up on it. You aren't correcting it till they've broken their focus and given up on the bad behavior. Another example would be people who try to put their animals off the furniture. They are lying there and the dog's all over them. They tell the dog to get off and it doesn't, so they push it off. It stands there and stares at them and the make it stay there for a while and then let it back up. They feel like they've accomplished something because they put the dog off and made it stay for a little while, but the whole time the dog was just waiting till it could get back on the couch. I would want to tell the dog to get off, then drive it off more than put it off. Sometime clapping the hands or flapping a blanket will work best, though with some dogs, this can encourage rough play and biting the blanket, which you don't want, so don't do this if you have this type of dog. Keep in mind that they more you claim a space and ask your dog to leave it, the easier it is in other situations, so if you've made a habit of asking your dog to move off, they will easily do so when asked to leave the couch.
A good alternative is to teach your dog the 'Spot' command. This is where the dog has a 'spot', such as a kennel with an open door, a mat, wherever. When you say 'Spot', the dog goes there. He can't be on the couch or jumping on you and be in his 'spot' too. I also wouldn't want the dog standing there staring at me, because he's still focused on what he wants. I will send him/her off if they do this. With repetition, you'll be able to say 'off' and they'll hop up and go find somewhere else that's comfortable for them. That's ideal for me.
So keeping all this in mind for your jumping issue. You'll want to make sure he doesn't have an abundance of energy to use to be destructive, you want to let him know that it's not okay to jump on or bite you. If he's persistent, you need to tell him 'no' and back him down. He can't just keep staring at you. He needs to give up on wanting to jump on you. Lastly, offer him an alternative to with a chew toy or puzzle toy to keep him occupied. Good luck and I hope this helped.