1. No you should see both linkshells and free company allocations on server update which i think is 15mins give or take.
2. If you mean Free company and that gives you access to more free company perks such as bigger company bank. never seen linkshell ranks on my game
3. Yes unique gear (often unsellable) comes in four flavors as far as we know green tint, pink tint, solid blue, solid purple they are arrange by rarity. Greens are uncommon and some are craft made, pink are drop specific (Dungeons) and have variable stats so better some worst, solid blue are quest rarities but only are used so far for artifact armor, and purple being the most rare which is used for relic weapons.
4. For specific rank I am unsure if it is just quest or if it is achievement based as well but for rep it is quests completed by members
5 Its either level passed or time since joining that restricts you.

Hope that helps and others clear up where i was unclear.