Can I see a transaction sent to me if I haven't downloaded the whole blockchain? Last received block was 12 weeks ago, and I'm still syncing. I have moved some bitcoins to this wallet from another, but I still can't see it in my inbox after a day.

Is it because I don't have all the blocks?
You can use blockchain.info to see transactions sent to your address.  Just go to https://blockchain.info/ and enter your address in the search field.