Thirteen is just a hyperactive version of Ten. Or, a toned down Eleven. Take your pick. ;)
That's quite alright! Natasha won't know who Gwen is, but she is familiar enough with their version of Peter.
I should warn you, right now the plot is a bit wonky. Everyone's trapped in 50s tropes (though it's totally optional if Gwen comes around just normal!).
Biggest thing is the town looks like Pleasantville.