You could use the the shower S/W to power fan!. Then via FSU as isolating S/W for fan, fuse rating for cable/fan.
Also best to put fan above/close to shower.
Agree about joist BUT all downlighters should be in Fireproof housing and these dont look like that ?.
Good idea about LED lights as they can save 80/90% of energy and 50,000 hrs, Plug and Forget !.
I usually use Osram as they have 15yr warranty.
Have fun.
You can get coated/plastic covered copper if required, best if burying in plaster OR Speedfit plastic pipe is best.
Have had no problems with either.
Always try pipework before covering just to be sure no leaks.
BUT may I suggest if your doing bathroom look at Plastic Cladding instaed of...