Why are signatures considered valid methods if identification when paying with a check or credit card?
Historically, we didn't have anything better. In theory, signatures are very good authentication methods, since hand musculature is relatively unique and it's quite difficult to forge someone's signature accurately without a lot of analysis and practice.
However, in practice, the people who check these signatures work in high-street stores. They aren't security people - they just want to process your order as quickly as possible and move onto the next customer. As such, even if your signature is only vaguely reminiscent of the one on the card, they will likely accept it.
On the plus side, many stores do store the stub with your signature on it, or send the stubs off to the bank for archiving. This means that you can go back and look at the receipt later if fraud is claimed.
Is this acceptable? From a security perspective, it isn't. It's not very strong, and in practice can easily be exploited. From a business and cost perspective, it's good enough and saves them a lot of money in terms of buying new equipment. The cost of fraud to the retailers is less than the cost of rolling out a better system, so they don't do it.
For that matter, what do those credit card readers do with my signature when I sign them? Are they validated or compared to my signature stored at the DMV?
I can't say I've had any experience of the DMV, but I've used similar systems at the bank. From what I can tell, they display the signature on-screen for the cashier, with some kind of overlay of the stored signature. It looks like it tries to point-match the two for a better idea of whether or not it is genuine. Arguably it could be defeated, but it does seem to be a better solution than a purely human validation. Of course, I'm also relatively sure that the cashier could accept or reject the signature regardless of what the computer shows - it's simply a visual aid.
One thing I have seen done, which does increase the strength of a signature as an authentication method, is directional matching. When you write your signature down, you do so in a very precise order. The direction of the loops and changes in acceleration / deceleration as you write are very difficult to forge. Observing a signature on paper and attempting to forge the way in which it was written is much harder than just forging the final product.
Extra points for people who can point to ways that companies are trying to better validate that I am actually authorizing payment for something.
Most countries (especially in Europe) now use chip + pin, which is significantly more secure and, as a bonus, much more eco-friendly! Less pens, less paper, less waste. I'm personally surprised that it's taking so long for the USA to adopt it.
Another new technology that has come out recently is contactless payments, which a few banks have adopted in the UK. It's a balance between the security of chip + pin, and the convenience of just using your card without any authentication. Essentially the card contains a small NFC radio, and you can make payments of up to £10 without entering your pin just by holding them against a payment system. There is usually a limit of 3 transactions before you have to enter your pin again.