- What is the purpose of this coating?
Generally it's to keep humidity or dirt away from components. This site has a good write up on it.
- Is conformal coating ever used to prevent service to a board?
Not that I know of, if people want to hamper efforts to service a product, usually potting is used, which is an epoxy that coats the board and all the components usually more than 1/4" thick. Very difficult to remove without collateral damage. Conformal coating can make it more difficult to service, but people that don't want parts serviced usually start by grinding off component ID's, then you have to etch off the package and find a nice microscope.
Is [the fibers in the conformal coating] likely that this will lead to problems?
Yes, if the fibers are not covered with conformal coating they could provide a pathway that penetrates the conformal coating. Ideally they should be removed before recoating, but this might not be possible. Acetone might be a better way to get the coating off, but whatever you use, be careful as some solvents can damage components.
- Is it advisable to re-apply the coating? If so: Are there any household products I could use for that? I was thinking about hairspray? Or is that a really bad idea?
Ha! Hairspray is not a conformal coating, I suppose they have the same consistency but are not even close to being chemically similar.
There are many places that sell conformal coatings, including digikey