If you bring a recent negative Covid-19 test result from Spain with you, yes (with the current rules). Because currently, people bringing a negative test result not older than 48h (from the time the probe was taken) are exempt from quarantine.

Without a negative test result, you are supposed to do a test at FRA and self quarantine until there is a negative result (for free tests, usually the next day). I doubt you will get caught if you fly away the next morning before a negative test result arrives, but in theory, you are supposed to self quarantine and not travel any further.

However, the rules are supposed to change soon and the ability to avoid quarantine by bringing a negative test result will be gone. Not sure when this is supposed to happen - first it was announced for October 1, then for mid-October, but it seems nothing has been decided yet.

I cannot find any specific regulations for transit passengers requiring a hotel stay. Maybe others can chime in.