So I lost a penny in it. My question then, that transaction fee, then goes to the next block? Or the one before the one I generated?
You didn't pay a transaction fee. You have .01000010 in your balance; you just can't spend it. The network would accept it if you sent the entire 0.01000010 to another address (you can't break it apart), but the Bitcoin interface doesn't support such high precision.
Since fees apply to transactions less than 0.01, adding a transaction fee should be at least 0.01 right?
That's not network-enforced. Transaction fees are not "outs", so they're not subject to the "dust spam" restriction.
it seems I paid the fee too. Why would that be? A block that had too many tx, the value coming from multiple small value purses (but the client should see one purse that satisfies the whole transaction, right?) or what else?
You received a fee of 0.01. You didn't pay one.