Logan is correct, but I'll go into more details about the first question:
You technically do not need to sync the whole blockchain, though you'll want to do it to save headaches. What you need to is sync enough to (1) get your owned outputs (or at least the ones which you will be sending), (2) get the transactions spending your outputs (or at least the ones spending the outouts you'd have tried to send otherwise), and (3) enough other outputs of matching amounts to fill up the ring signtures (to meet the minimum mixin).
If you happen to have those data, then you are able to spend your outputs just fine. This may seem to be pedantic, but it may be the way to make a lightweight client at some point.
This is wrong - this will reduce the time that simplewallet takes to scan the blockchain, it does not address the original question of whether a user has to download the entire blockchain. For instance, this part of the original question "If I know the block-height when I received my first Monero payment, is there a way I can save time by synchronizing only past that point?" has to do with a block height on the blockchain for synchronizing the blockchain.
– Ginger Ale Jul 20 '16 at 09:56