1. Right now, at official tournaments, there is only one format, called "Modified". What this format is composed of changes from year to year. Currently, it consists of the sets listed here:
https://www.pokecommunity.com/threads/309435 . It's due to change in August after the World Championships, but the changes will probably be announced sooner than then. Starting in August, though, there will be a format that starts at the Black & White set and continues onward to the current sets running alongside the current Modified format at events.
Casual play is frequently referred to as the "Unlimited" format, which includes all cards that do not say that they are not fit for play. It frequently is left up to "house" rules on what cards are allowed, though, since there are few if any officially sponsored events run using this "unlimited" format.
2. See my first paragraph of my answer to question 1.
3. TrollandToad, top-rated eBay sellers, and the odd local shop that deals in them AND has a league.
4. Not that I know of, but I don't really use Facebook at all for those purposes.
5. Tournament results are found somewhere on this site:
https://www.pokemon.com/us/play-pokemon/pokemon-events/ , though they don't usually release full player names, and the site did change recently, so I'm not sure exactly where on there you find that information, just that it's somewhere on that site. Decklists used at tournaments tend to not be published by Play!Pokemon, other than those used by Worlds Winners. Creators of tournament-grade decklists tend to write articles for the various TCG competitive play news sites like SixPrizes, if they even choose to publish their lists at all. Most top-tier players do not post their lists online to reduce instances of people net-decking their decks.
6. There are several "tiers" to rarity in the TCG, and it varies from set to set. Past Rare-holo in my list below, the rarities are just a rough approximation of that card type's rarity:
Common - > Cards with the circle on them. These are the most common cards. They often come in alternate/reverse holo prints, but those are quite common as well.
Uncommon -> Cards with the diamond on them Also come in reverse holo prints.
Rare -> Cards with a star on them and no other special markings, except that many have reverse holo prints.
Rare-holo -> Cards with a star on them and a holofoil picture with no special markings on it. Can have reverse holo prints in most cases.
Ultra-rare -> Cards like Lv. X cards, EXs,
exs, Primes, etc. Have a star on them and are special card types.
Ultra-rare level 2 -> Full art versions of EXs. Noted by a star and have a collector's number within the limits of the set's "card count".
Secret Rare - > Cards like Shining Gyarados and Espeon* that have a collector's number outside of the set's "Card count".
This help at all?