Hi,
Regarding your questions yes I've never seen a watch being serviced without removing the strap.
On the second point your watch can get magnetized instantly if placed where it should not (near some computers or Tv sets for instance, or other electrical stuff).
If your watch is slow one minute per month that means two seconds per day. That's not bad at all. Personaly I don't like slow watches, I prefer my watches to be (if necessary
) slightly fast.
You may try to go to the JLC boutique and ask them to regulate accordingly. It can be done fastly. They can also put the strap back like you want.
Cheers
Dje