1. Correct, that is the general view. The main advantages are 1) that it (mostly) removes disagreements about the coin's grade out of the negotiation process because the coin has already been examined by experts; 2) it guarantees authenticity – though, nowadays, slabs are getting faked too, especially when it comes to older Chinese coins; 3) it prevents damage from mishandling the coin; and, yes 4) slabbed coins sell for more because of the above factors. The disadvantage is that you can't hold/feel the coin in your hand. There are purists that will go as far as break coins out of any slabs they buy.
2. As far as I know, there is no standard. They're just big in Europe, so a lot of people there use them. There are other brands available in the US. All you really care about is that the capsules are airtight (to prevent the air from getting in and oxidizing the coin's surface) and the plastic doesn't contain any chemicals that might be harmful to the coin / cause PVC damage.
3. If you want to go for a professional setup, use a camera with a macro lens + tripod + build a light box. Though, you can get pretty decent pics even with a phone camera. Consider buying a digital microscope if you want to photograph really small details. They're pretty affordable on E-Bay or you can buy a fancier one from Lighthouse. Personally, I hate photographing coins, especially proof ones, because of how hard it is to capture luster and get a well-focused image and same-sized pics of both sides. Flatbed scanners produce images that look flat and fail to capture any luster.
4. Capsules for proof coins, 2x2 cardboard holders for everything else is usually sufficient. For 2x2s, either self-sealing or ones that you staple together is fine. Self-sealing tend to be thinner (which maybe a factor if you put them in plastic pages), but they usually cost 3 times as much. If you go with stapled, I would suggest buying a flat-clinch stapler. You can store the 2x2 coin holders either in a cardboard box or plastic pages (usually 4x5 pockets per page) that fit into a 3-ring binder. Lighthouse sells all of these supplies. In the past, people used to store coins in little paper envelopes, but that has mostly gone out of fashion, since coin surface does react with the paper and causes toning. Some high end collectors use wooden munz cabinets for silver coins, so coins can sit in the open and develop a “natural” patina, but that's a matter of preference and a pretty expensive option and won't be necessary for a novice collector.
Though, considering what you collect, wooden trays might be your best option, since hammer-struck coins might be thicker and not even fit into a 2x2. Plus, they have already been exposed to air for hundreds of years and developed a patina that protects them. Then you would only need to worry if you live in a moist climate or have smokers in the house.
