Yes. You can export your key from the app if you want to.
You don't need to. Ulys works without it, and your wallet is always recoverable by signing back in — there's nothing to write down or store to keep using Ulys normally. Exporting is there for people who want their own copy.
We can't do it for you. Ulys can't view or export your key. That's what makes it uniquely yours, and it's also why we can't recover an exported key if you lose it.
If you do export it, treat it the way you'd treat the key to a safe: anyone who has it can move your funds, and it can't be cancelled or reissued.