Yeah, a fifteen-fret uke will have two octaves. The lowest note is middle C. Then the E string (open) is the E above middle C, the A (also open) is the A above middle C, then fifteen frets down you have the second C above middle C. That's two octaves.
Yes, peanuts and peanut butter can aggravate arthritis. I was a peanut butter eater all my life, I am now 58.
Peanuts have protein in them which is good but they also have satuated fat which is not good It would not hurt her arthrtis at all but they may affect her BP if they are salted because the salt will cause her to retain water and in turn the water weight will drive the BP up.