Practically speaking, couldn't I just tweet the SHA1 hash and let Twitter be my trusted timestamp authority? Assuming I don't work for Twitter, it's unlikely I'm able to modify the timestamp on a tweet. (The blockchain has a lot more confidence due to its distributed nature, but in the minds of say, a jury of your peers, I bet it actually appears shadier than Twitter.)

Yes, but that would be a centralized solution. When Twitter dies (wont happen, but still) or a rogue admin changes the timestamp, etc. - It requires a trusted party - Twitter - instead of the blockchain.

For a more permanent timestamp, just post your hash in a newspaper's classified ads.

Also post it on your Facebook, for prosperity.

And, even if that doesn't somehow make you rich, at least you've got it there for posterity, as well :-)

What risk does that pose in this scheme?

Wouldn't it be possible to forge the SHA1 or a document?

Hash extension attack yes, but only if you already have a valid hash and document and wish to add more to the end of your document. This however will change the hash.

I also meant to type SHA2, but it would make no difference since you can still do length extension with it. In fact, you can do length extension with anything that uses Merkle-Damgard.

You should use Sha256. Then, the answer is "No" to both questions.

This is a horrible idea. DO NOT DO THIS. You are making the bitcoin network worse if you do.