Do sincere repentance and abundant supererogatory prayers obviate the need to repeat past prayers that were performed late or with tayammum when water was available?

Your Fajr prayer, if performed after its prescribed time, is valid as a make-up prayer (qada'). As for your previous prayers performed with tayammum while you were able to perform wudu', if you were unaware of the ruling, then you may act according to the opinion of those who do not obligate repeating them. However, if you were aware of the ruling, then the majority of scholars obligate you to make them up. You should investigate and make up those prayers until you achieve certainty or strong presumption that you have fulfilled your obligation. And increase your acts of seeking forgiveness (istighfar) and performing good deeds that expiate sins.