Yes, that was likely the handwriting in vogue at the time. It is beautiful, however!

I don't know what kind of ink they would've used if not iron gall. Also, modern IG formulations differ from older ones, I think.

As far as the handwriting goes, while the one from the note is recognizable, the script you learned at school was likely different. Ref: https://en.wikipedia.org/wiki/Teaching_script

I was taught https://en.m.wikipedia.org/wiki/D%27Nealian

The uppercase t and h are the two main differences, but I guess I adopted the spencerian or copperplate version at some point.

Lampblack ink or india ink was in use as well and spots with lots of pigment can have it move when wet.

Edit: it would seem synthetic aniline based black inks were in use for personal writing by that time. Aniline black isn't even close to waterfast, so that might very well be it.