Actually you can mix and match IOS versions and IOS feature sets in 3750 stack. What must match, I recall, though, is the major stack version being used by the two IOS versions. It's also good if the minor stack version matches too. The easiest way to insure that, is to use the same IOS version on all stack members.
As to auto upgrade, that I also recall has its own set of rules and restrictions, and it was something I never used, beyond perhaps having IOSs with different feature sets within the same stack.
As to copying one switch's IOS to the other, I believe that physically possible from either to either. Legally, though, it's a different matter especially as your one image is IP Base and the other is IP Services. It would be a license violation to install the IP Services images on the 3750 without a IP Services license. (BTW, interestingly, you can run IP Services on the stack master, which would effectively have the IP Base switch have IP Service features in a stack, as the master image controls the whole stack. [If the IP Service stack master fails, the remaining switch would revert to IP Base.])
I recall originally the 3750 series had a lifetime software upgrade license, but I also recall, Cisco later changed that. I.e. It might also not be legal to upgrade your IOS version on either switch without paying a license fee.