Situation: I want to run a full node to do my small part in the distributed Bitcoin network. I have Block Erupters and I'm willing to run them solo mining despite not having any chance of finding blocks if it really could help the network process transactions.
Question: Since I would never solve a block would the transactions I process really contribute to the network?
If the answer is NO then why run the Satoshi client? Is it just to distribute the Blockchain?
No, transaction confirms ONLY when a block is found. You can contribute more by joining a pool although you have a slim chance of finding a block solo mining at low hashrate.
Satoshi client is only helpful if the port 8332 is opened. Satoshi client helps to relay transaction to other nodes. It also distributes block to other clients and helps other who are new or who are blocks behind to download the blocks from them. Satoshi client is only helpful if you run the client 24/7, it is recommended to run light weight client if you don't keep your computer on at all times.