Did the battery pull itself fix the sms issue? Be sure to pull it out with the phone on
How many emails could you have gotten in a week that would make your phone lag? I know if you keep too many too long it can cause issues, but I don't think that is the issue because you have had it a short time.
You should get your phone swapped, it could be defective. Since one of the keys is not working and you are having general hardware issues, you may want to check the stores policy on swaps. Make sure you backup first before doing so though!