Is it possible to trade BTC and other altcoins on Ethereum's blockchain? If so, how?
it is. You want BTCRelay. They spoke for a bit at devcon. It's essentially a means to utilize smart contracts to deposit and trade bitcoin.
The main issue is that you can not create a smart contract that will interact with a BTC node directly. None that I know of anyway.
From ETH point of view BTC is just like Gold or USD or EUR. It is an external value store over which ETH has no control and can not interact with directly. You can create a BTC backed coin by buying BTC and promising a fixed exchange rate. Just as you can with USD or Gold or the SDR. Which are things that exist.
The only way I can think of is to make use of a trusted 3rd party between the two wallets (ETH and BTC). A program which is able to interact with both ETH and BTC. This trusted 3rd party can also be a person it makes no real difference. But a program is faster to react to requests.
You can verify on the ETH and BTC blockchains that the 3rd party did what it claims to have done (unlike with Gold or USD). But the problem remains that this 3rd party can do whatever it wants (or is programmed to do)
The short answer in that case is NO.