Should I re-apply conformal coating after hot air soldering?

Conformal coating is used to increase mechanical toughness (like abrasion), shock dampening, and to protect against moisture, dirt, grime, and arcing.

Certain types of conformal coating are very nasty to get off if you need re work so they can be used to prevent service. But normally you would just use epoxy for that.

Silicone conformal coatings can be soldered through for rework.

Note if conformal coating is applied improperly it can do more harm than good. Silicone, for example does prevent moisture from getting to the PCB, but it is not completely impermeable. Therefore, if air pockets exist, the moisture can permeate through and accumulate against the PCB where they it will never leave.

Do NOT coat a board with anything that it is not meant to be coated with. Hairspray, for example, is meant to stick and stiffen things, not seal them off. Furthermore, isn't hairspray flammable?

The most common types of conformal coating that can be applied by a normal person is acrylic, polyurethane, and silicone. I prefer silicone for reworkability. It's feels softish and pliable. The others are nasty get off.

Paraylene is the really nice one but requires vacuum deposition equipment so it's NASA/military stuff.

1) 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.

2) 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.

4) 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