Are you asking if the email could be attributed to someone strongly enough to be used as evidence? As in, if I threatened to kill you in email, would you be able to have me arrested?
Or do you want to know, assuming the email is strongly correlated with the supposed sender, is that enough to make a valid contract.
As for #1, email is pretty hard to forge to make it look like it comes (to the trained eye) from someone specific. It’s trivial to forge untraceable email. If the email originated from your IP address, and your ISP had logs of it (they do), then it would be pretty good proof.
As for the acceptance… Hmmm. I know it could be enough. I get a lot of contracts via email and get paid on completion, but then these people are long-term clients. I’m pretty sure you have to reply via the same means, or in person, to make a contract valid. If I called you to offer, you’d have to call back or talk to me in person, to accept. (Or, to be able to force me to accept your acceptance… if I want you to accept, I wouldn’t fight it.)
But, assuming the email is proven to be from both parties, and you offered and I accepted via email… I think it would be binding. I think you I might have problems forcing you to accept, if for instance, you offer me a contract, then change your mind, but before you tell me, I accept. I think I’d have a hard time forcing you. But, once I accept, especially if you acknowledge, I think it’d be valid. If you entered into the contract knowing that it would be invalid, or intending to break it, I think you could probably be charged with fraud.
But, the example you used doesn’t appear to be a contract. You didn’t get anything for doing this, so it appears to be a friendly offer, not a contract. In a contract both parties must get at least token consideration, hence the cliched $1.
As for really determining ID, if I sign a message with PGP or any similar public key crypto, it’s astronomically unlikely (literally, 10^50th kind of thing) that you could have forged that. You’d have to have stolen my key somehow. So, it’s even better evidence. But I don’t know if digital signatures are yet recognized by US/Canadian courts.
So, mostly guessed, but I hope it helps a bit.