1) Most classic achievements aren't going anywhere with Cataclysm. Of those you cited, none are going anywhere, though the requirements for some are changing (If you already have them, you don't need to worry about it, but if you don't, you'll need to meet the new requirements. In some cases, these are actually easier than the old ones - Zul'Gurub is being removed from Classic Raider - in others, it's essentially resetting all your hard work on, for instance, Loremaster, if you don't finish. Some other changes:
- Did Somebody Order a Knuckle Sandwich and Master of Arms were changed to Feats of Strength in Patch 4.0, with the removal of Weapon skill. If you missed out on these, it's too late already, sorry!
- The Green Hills of Stranglethorn, and with it, the collected Nesingwary achievements, are - somewhat surprisingly - not going anywhere.
- With the removal of Zul'Gurub, The Deadliest Catch - will become a FoS, and no longer obtainable. Similarly, if you have any interest in a rare Raptor or Tiger mount, with it's attendant FoS, you have about a month left to farm one up.
- Zul'Gurub is also where you go to farm rep with the Zandalar Tribe, whose reputation is being removed with the Cataclysm. You can also buy this rep off of the AH in the form of a whole lot of token turn-ins.
- Since you asked - the Hydraxian Waterlords, (and Molten Core), aren't going anywhere. You're welcome to keep farming that rep well into the expansion.
- Finally, while Insane in the Membrane is not being explicitly removed, several of it's requirements will become impossible after the Cataclysm. Notably, there will no longer be any way to gain Bloodsail Buccaneers Rep (which also means if you want that Bloodsail Admiral title, you're running out of time for that too...), and the Librams needed for Shen'dralar rep no longer drop anywhere, and the NPC's to turn some of them into have disappeared. Ravenholdt will still be doable, but significantly more grindy, as the Syndicate mobs which you used to be able to kill to get to Honored have been removed - it's Junkboxes all the way to Exalted.
2) No. When the Shattering happens, it happens for everybody, regardless of their sub status. The world changes will be reflected in the vanilla game, and not buying the expansion is not going to buy you time to work on old achievements. The clock is ticking!
3) There are no TBC achievements changing as far as I know, but several Wrath raiding achievements are going to get the FoS treatment, since attempting them at level 85 trivializes them. These include the various ToGC tribute runs, The Immortal/The Undying, Champion/Conqueror of Ulduar and He Feeds on Your Tears
An easy way to check on future availability of any achievement in which you're interested is to look it up on WoWHead, and then click on the "Cata" link in the top left corner, which will show you how it is being changed in the Expansion (if at all). If it's been recategorized as an FoS, or had it's criteria changed, you should see the difference.
This question gets a bit tricky to answer... Starting from scratch, if you buy just vanilla WoW, you can experience the original world as it has been destroyed by the Cataclysm expansion. You can level up to level 60, but cannot access professions, areas, races, or other such things from any of the other expansions.
In order to access Outland, jewelcrafting profession, and Draenei? / Blood Elf races and starting areas, and level 61-70, you will need vanilla WoW + the Burning Crusade expansion.
In order to access Northrend, Inscription profession, and Death Knight class, and level 71-80, you will need vanilla Wow, Burning Crusade expansion, and Wrath of the Lich King expansion. Oh, and a prior level 55+ character on your account to play as a Death Knight.
In order to access the misc Cataclysm level 80+ areas, Archeology secondary profession, Worgen / Goblin races, and level 81-85, you will need vanilla WoW, Burning Crusade expanion, Wrath of the Lich King expansion, and the Cataclysm expansion.
Those are the restrictions. You can play on whatever realm you want with whoever you wish. You just may not be able to reach all the areas they can.
You can trial every bit of it with the 10 day trial, though I don't quite know how they treat it when your time is up.
Best Answer
Achievements for vanilla WoW appear under the "legacy" tab in the achievements list. Most of them were retroactively awarded to players once the achievement system was implemented.
I have not been able to find a full list of legacy achievements specific to vanilla. However, you can find vanilla achievements easily enough by searching the wowhead database for legacy achievements with "60" or "classic" in their description. I cannot guarantee that all of them have this identifier though.
Below are links to all of the legacy achievement lists on WoWhead:
General
Dungeons
Raids
Professions
Player vs. Player
Currency
Expansion Features
Scroll through these or use Ctrl+F and look for "60" or "classic". Off the top of my head, I know you'll find some vanilla achievements related to pvp, dungeons, and raids. For example, if you browse the legacy PvP achievements and search for "60", you will find achievements that rewarded titles based on the highest achieved PvP rank during vanilla, such as Blood Guard. There are also some dungeon and raid achievements, such as Onyxia.
Note: As mentioned in the comments, some of these achievements were actually attainable after Vanilla.