Question is - can I restore from SQL 2012 SP1 to 2012 SP3 and use standby mode? Or is that counted as a different version?
Yes you can restore the user database backup and this would be possible because version remains same, although this is NOT the recommended approach. Ideally the service pack level should be same. After you have done the restore please plan to upgrade the primary server to SP3 ASAP.
And if so, can I log ship from enterprise to standard for example?
Yes, subject to condition the backup does not includes enterprise features data or the time you took backup of enterprise DB it should not be using any enterprise features other wise restore on standard will fail. To see whether enterprise features are used query enterprise DB with
select * from sys.dm_db_persisted_sku_features