How do I reward a miner who made a block I like?
It seems obvious that you could just send some coin to the receiving address in the coinbase transaction, but usually, that address is for a large group of people.


For pools that payout via the generation transaction (cf. this one), it is not one address "for a large group of people," so you could not "send some coin to the receiving address" because there's not just one. (Generation transactions have no inputs by definition.)

Check with the particular pool (e.g., Eligius pool lists the winning miner in a column entitled "Contributor" here). Oftentimes they say who mined the block, and you can send them BTC directly.

Not all pools pay the miners via the generation transaction, but some do (Eligius and P2Pool were a few of the first to).


  When you're in a mining pool, if yours is the machine that finds the solution to a block, then you get a much larger reward than everyone else.


That's up to the rules of the particular mining pool. See this comparison of the different pools' reward strategies.