I mine at Eligius pool and sometimes a new transaction appears in my bitcoin client labeled 'Generated'. Why does this happen? Do I also get these 'Generated' bitcoins if I use an e-wallet's address for mining?
The special ‘coinbase’ transaction, which begins every block of the main chain, usually contains a single output with a single address (owned by the person or the pool that generated the transaction).
However, the fact that this transaction usually contains a single output is not enforced by the system. You may have a look at what an Eligius block looks like. The Eligius pool uses this to credit its users directly.
It's not possible¹ to include inputs in these transactions (the block would be invalid otherwise). The only special input is the 25BTC one (currently, every 4 years this value will decrease) granted to the block founder.
return (vin.size() == 1 && vin.prevout.IsNull());
Generated coins mean you are getting new bitcoins that were generated with the block. Eligius has the pool setup to pay you out when you have over a certain amount of work put in at an address. Rather than having all 50 coins go to his address and then pay you out, he pays everyone out directly by having the coins paid out upon generation.
For the second part of your question:
Some EWallets can handle generated bitcoins, some can't -- depending on what client they are using. If the wallet refuses to support these, you could losse funds. i.e., it is best to not use an EWallet when mining on Eligius, P2Pool or any other pool that generates coins directly to the address supplied by the miner.