This doesn't affect myetherwallet, does it?
I think the issue will be resolved as soon as possible, but then its impact and people's trust will go very low.
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Can ethereum classic do the same things that ethereum does - the smart contracts and all that...?
No it doesnt. It only has to do with Parity wallet which is not something good at all as it affects the whole Eth community.
It could happen to anyone as long as you are holding ETH. This is one of the reasons why Waves will start being the next best option very soon with this sort of vulnerabilities. This is the second time something like this will happen with Parity.