bitcoin-qt does not understand stratum. So Slush mining proxy is of no use. It implements the getwork protocol. I guess you can say that it acts like a mini-pool but it does not scale very well.
It understands getwork and GBT (getblocktemplate). There is talk about removing getwork. Pools use GBT to talk to bitcoind. I haven't tried but you could probably connect a mining client using GBT directly to it.
Is there any way to figure out how much I earned for each block via the API.
Not currently. I'll think about that for a future update. I am planning to expand the API quite a bit. All ideas are welcome.
is there any way to see in Transaction history the summary of earnings?
That's basically the current balance, or did you mean something else?
Each transaction on your account has a category, even though it's not shown on the website yet. I am planning to show totals by category. So you could see how much your total mining income is, how much you donated in total, etc.