Can I create a new key pair for my SSL certificate without getting the cert reissued?
Nope. You can't do that.
If you've lost the key, you'll have to get it reissued. Most certificate providers are OK about this, though. Some even offer insurance for it.
I've had a certificate regenerated about 4 hours after I had it issued, because i got the CN wrong.
The certificate contains the digital signature of the certificate authority. This signature certifies that the SSL certificate was not changed. The certificate contains the public key too and this is generated from the same prime number as the private key (for RSA certificates). So it is not possible to change the private key without changing the public key and in turn without changing the certificate. Changing the certificate will render it invalid because the signature will not match the rest of the data.
Here is the procedure for creating a new certificate if the old one is compromised:
If security is the top priority do step 5 first.