"We save the mother on time" suggests a certain deviousness. It suggests planning. "On time" could mean "on schedule, as planned, as intended". That works with constructions like "The bus/train left on time" or "The play started (right) on time".
"In time" implies "before the opportunity to do so was wasted/unavailing/useless".
"We arrived in time" suggests "before it was too late". "We arrived on time" suggests "at the right time to participate".
But that would need different cases: "We save time for the mother " (i.e., we save time for her benefit): time would be accusative, WHAT we save ( = Latin tempus); mother would be dative, TO/FOR WHOM we do the action (Latin mātrī).
"Time" is a 'weird' word, being a neuter of the 3rd decl: tempus, temporis , neuter.
No--since tempus is a 3rd declension noun, its genitive is actually temporis . (It's neuter, like corpus, corporis , n., body; not a typical "us" noun like Marcus or servus belonging to the 2nd declension--those are the ones with genitive in -ī .)
If it's a case form, it's the dative singular. Sometimes nouns seem to have developed what are essentially adverbial forms, and this may be one of them.
(For example, you can say "at night" either with the ablative nocte , from nox, noctis , f., night; or with adverbial noctū , as if there had been a 4th decl. version of the noun.)
Two different verbs:
servāre means "to save, preserve, conserve," and the a of servāmus shows that the verb form in this sentence comes from servāre : "we save."
servīre + dative means "to serve, be a slave to," and the present-tense "we" form of it would be servīmus .