Does rinsing the nose by putting water in the palm of the hand then placing it inside the nose and expelling it, not by drawing water in with the breath, make the ablution and previous prayers valid?

Your previous ablutions and prayers are valid, because rinsing the mouth (madmadah) and sniffing water into the nose (istinshaq) are among the Sunan of ablution, not its obligatory acts, according to the majority of scholars. Consequently, whoever omits them does not have their prayer invalidated.