I always just used a paper towel to wipe the processor down first, then used qtips dipped in rubbing alcohol to finish it off. All while leaving the processor in the laptop.
Just don't do something stupid like pooring the cleaning solution directly on the CPU :-P
As for the thermal compound, you really don't need a lot. Just put a small dot on the processor and then spread it around with whatever you are going to use (small piece of cardboard cut from a cereal box works great).
The instructions for the ArctiClean are to put a few drops of the solution onto the die of the CPU and it will then emulsify any old compound that's there so that it can easily be wiped off.
So potentially there is a risk there of the solution dripping off the die and onto the motherboard.
Not sure if that solution would then damage the board.
Do you think it's better to spread the compound before putting the heat sink back on or is it better to let the pressure of the heat sink spread the compound when you put it back together?
Anyways, using a q-tip will still spread the cleaning solution over the compound causing it to emulsify. If you want to put it directly on then I suggest buying an eye-dropper which can be bought at various stores like a pet store or pharmacy. Even if a tiny bit gets on the motherboard, clean it up and thoroughly let it dry and you should have no problem.
As for the compound, since it is non-conductive it is ok to get on the pins, but you really don't want to. You shouldn't be putting so much on that lots of it is spilling over the sides. I don't subscribe to putting one big glop on and letting the heatsink spread it since it does not guarantee 100% coverage (it could all come out one side). Even if you spread a thin layer over the CPU it will be enough to fill in all the irregularities between the CPU and heatsink and it allows you to make sure you aren't having a lot spill over the sides. You can even leave a small (~1mm) edge around the CPU to be covered by the heatsink spreading the compound.
The Arcticlean solutions are in bottles like you'd get eye drops in so that should be ok for controlling the amount I put on. It's probably a good idea though to just put a couple of drops on and then spread it around with a cotton bud rather than trying to drip the solution on all the grease it needs to dissolve then
If I do get something on the board, what's the best way to clean it off without causing damage? Would trying to use a cloth or something like that introduce static to the board?