At last I solved it! From the perspective of someone who doesn't usually do crosswords: this was quite difficult. I did have to resort to Wikipedia-ing and (in some way worse) Googling for many of them, but it was fun.
ROT13 of my solutions that the website in the original post will accept as input to the "Save/Load" button after decoding:
...NJR ZON EGY QBTR NFPU/SNA BEQ NBV NANF PUNV/NTV YVM FBA OBYGMZNAA/ERQOBAR ZYX OUB ZBQ/ HPT GHN FVZVRFDHR /CREGU ONFVYVFX GRH /RYNY EZF RZZL RENTBA/REARF RER QHNY ABAB/GVNEN RVA YVR N
a) This is really cool.
b) I recognised "TheNitromeFan" - turns out I'd seen them on Reddit as u/TheNitromeFan. Interesting.