Where do these coins gain their value? I have been mining coins for a long time (started with BTC/LTC years ago, moved to altcoins recently) and have realized I am not clear on the actual source of the value.

Do the exchanges deem the value of these coins? With all of these new alt-coins, do they not actually have a BTC/USD value unless they are added to an exchange?

How do buy/sell orders affect the coin price? If someone bought X amount of , that would cause the value to increase correct? Wouldn't they then just be able to turn around and resell what they just bought plus any they already had for a higher value?
This is really asking about basic trading economics: if I have some new product X, what price does it have? I can set a price and see if anyone buys it. If not, I might lower the price or auction it to see what someone might pay for it. That is the starting value, but it doesn't really mean much as it hasn't been field-tested (so to speak). If others think it is worth more, they will buy more; if others think it is worth less, they will not buy at the current price, so those trying to sell will lower their prices until they can agree on a price.

What you describe is a simple pump and dump operation, which could be done on any sort of traded commodity like stocks or cryptocurrencies (especially altcoins). If you can feign interest in something (in this case, an altcoin) so that the trade price goes up, then you might be able to turn a profit. Note, however, that this is a sort of fraud and can open you up to, at minimum, civil lawsuits.

Doing a "pump and dump" will be easier on small markets, like altcoins, than on larger markets, like Bitcoin and Litecoin, because you have to buy up enough of the coin to start an upward trend that people will notice and buy into. If the market is large, you'd need huge resources in order to make an impact in the market.