When mining with 2 computers, do I need to use 2 separate wallets addresses? I understand that each miner is working on a different block. Correct me if I'm wrong but this is because the first transaction in a block basically contains my wallet address so I can receive the bonus of solving the block. Obviously everyone wants to receive the bonus so that first transaction will be pointing to everyone's separate wallet addresses.

Assume that I have two computers solo mining with both pointing to my wallet address, wouldn't this mean my computers are solving the same block? Continuing with this train of thought, wouldn't that mean that my slower computer is basically useless because it would be generating the same hashes as my faster computer?

Now how does this apply to pooled mining? In a pooled community, is everyone solving the same block which has the first transaction pointing to the pool owner who then distributes the coins?
Regarding pools: when working for a pool, the computer who solves the block doesn't credit its owner directly - it rather credits the pool's bitcoin address. The pool manager then tallies how much work each of the pooled miners delivered for the block, and proceeds to pay them proportionally to it - minus the fee the pool takes, of course.

The only exception is a P2Pool, which work with no centralized pool manager. In these type of pools, each of the peers in the pool ensure they're appropriately credited for their work when a candidate block solution arises.

As far as I know, you can configure your 2 solo-mining computers to either use each their own address or one for both. And no: even if both were using their own addresses and working independently, the processing power of the slower one is not completely wasted. Solving a block is highly random, there's not much chance both would be trying the exact same nonces in sequence. Rather, they'd each try a different set of nonces to solve the block. Obviously, in the long run the faster one will find answers more frequently than the slower one.