LP Token is just one name for the tokens you get, when you provide liquidity to a pool/pair, they may have other names, depending on where you read about them.
Uniswap tokens are something different, they allow you to vote on some things related to the future of Uniswap. They are listed at Coinmarketcap (currently #8) and may be traded at some Exchanges.
The LP tokens you get, when you provide liquidity to a pool/pair, can always be burned for your share of the both ERC20 tokens of the pair, so there is another LP token for every pair.
Because it is exactly clear what you can get for it, there is not really a need for them to be traded somewhere, because their price is perfectly defined by the amount of ERC20 tokens that are in the pool right now.
So LP tokens would not be used by traders, because they can trade the ERC20 tokens, the pair is build on, directly.
There might be a own pool/pair, where you can buy or sell them directly for some LP tokens, because they are ERC20 tokens as well and nothing stops someone from setting up a pool for them (I don't know if there are, but it is technically possible).
When you deposit liquidity to a pool you should be careful to add both tokens around the same ratio as they are in the pool at that time, because otherwise you will get far less LP tokens (down to nothing in v2, when you only add one of the tokens and nothing of the other).
The formula (directly for the soucre code) for the amount of LP tokens you get is
Math.min(amount0.mul(_totalSupply) / _reserve0, amount1.mul(_totalSupply) / _reserve1);
where amount0/1 is what you deposit of token 0/1, _reserve0/1 is how many of token0/1 are in the pool right now and _totalSupply is how many LP tokens there are for the pair right now.
No, Uniswap tokens are not LP tokens, there is another LP token for every pair.
Providing liquidity is the main way you get LP tokens, for some LP tokens there might (technically possible) be a place (propably another Uniswap pool) where you can trade them. It is unlikely that they will be traded at a large central exchange.