While we've not done a gluten free kit, I'll try and answer what I can based on other beer kits, but other people might have different experiences/advice.
1: We use just boiled water from a kettle. It's pretty hot to dissolve the malt syrups, where cold or warm water will be much much harder to mix.
2: It will depend what your water is like in the first place. If you filter it to drink a glass of water, then I'd say use a filer or bottled water for your brew. If your water is pretty good anyway shouldn't be too much of a necessity. We've done kits using supermarket value water, which worked out about 10p a liter. We then used those empty water bottles to bottle some of the brew too, but I wouldn't suggest them for long term storage.
3: Hmm, honestly can't advise on this one, but if they recommend it I'd follow their experience.
4a: Finings can be a good idea to aid with clarity of your beer. But time and gravity will do the same job, only slower.
4b: Yes, stirring it into the beer would mix in all the sediment. Many people will rack the brew into a separate container before mixing the priming sugar in so that this doesn't happen. Alternately you could add the priming sugar evenly into the bottles, or use carbonation tablets, eg.
http://www.tesco.com/direct/coopers-carbonation-drops/213-3925.prd
We've started to bottle more of our brews now, giving us more choice. We've got a dedicated bottling "bucket", with a bottling tap, so we can rack the beer into this off the yeast bed, and then batch prime (add the priming sugar to the whole batch of beer) before bottling. For this we picked up a cheap fermenter from Wilkos, reduced where someone had stolen the lid and added a "Little Bottler" (
http://www.brewbitz.com/Little-Bottler__p-1068.aspx?gclid=CIubivrOncUCFRHLtAodxn8A9Q).