If I have a license of SQL Server, can I install it on a normal Windows Amazon EC2?
Will data I save in an EC2's instance be saved if I turn it off and start it a day later?
It depends on what your license says really. Sometimes they're very specific about putting only on servers you "own", but if you were previously putting it on rented servers then this is no different.
EC2 data is not preserved if you terminate the instance, however you can either use an EBS volume to store your data (which does persist), or create a snapshot AMI whenever you want to completely preserve your server.
EBS backups are good for backing up files, while an AMI is better for backing up your entire server config. Doing both wouldn't kill you either.