I’ve long suspected that the First Doctor wasn’t on his first regeneration cycle. We’ve on screen evidence for this but was never followed up.
There's only one piece of "evidence" for that, the suggestion of multiple earlier Doctors in "The Brain of Morbius," and it conflicts with other references like "The Three Doctors" having the Time Lords explicitly refer to Hartnell as "the earliest Doctor" and "The Five Doctors" (and "Twice Upon a Time") calling him "the original, you might say." Everything except "Morbius" has always consistently presented Hartnell as the very first incarnation of the Doctor.
Of course, Who continuity has always been made up as it went. The whole plot of "The Deadly Assassin" depended on the idea that it was impossible to give a Time Lord a new regeneration cycle without destroying Gallifrey and half the universe, but then "The Five Doctors" came along and the Time Lords were offering the Master a new cycle as if it were no big deal. Even so, "The Time of the Doctor" established that, while it was possible to restart a Time Lord's regeneration cycle, it was exceptionally rare and only done in extraordinary circumstances. The Doctor fully expected to die, and it took all of Clara's persuasion to convince the Time Lords to do this special, unprecedented thing and give him a new life cycle. I think it would cheapen that horribly if it were retconned that the same thing had already happened to the Doctor once before.
Besides, I got tired of Moffat always making the Doctor the most exceptional and important and special person in the universe. I don't want Chibnall to perpetuate that bad habit of making everything in the universe revolve around the Doctor. I can accept the Doctor earning a second life cycle because it's necessary to keep the story going, but revealing that they've gotten an extra life cycle twice already would just be obnoxious. Stop making everything about the Doctor. Let the Doctor just be a traveller helping people out, like she said.
Thirteen mentioned last week that it was a long time since she’d bought women’s clothes, though the doctor usually steals them.
He could've bought clothes for Susan when she was enrolled in Coal Hill School. He was trying to blend in and avoid attention, so buying clothes would've been preferable to stealing them. And didn't he spend a fair amount of time living with River in the months before the end of her life? He probably bought her clothes as a gift at some point.