World-of-warcraft – How big is the update for Cataclysm after buying the digital upgrade

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.

If you are starting from scratch and no patches, then the Cataclysm DVD should bring you up to 4.0.3a. They sent out many Warnings not to install the previous games on top of patch 4.x, so it stands to reason that they would just give you the full game on Cataclysm as they have in the past. However there is a topic over at mmo-champion indicating that the game client might try to download the BC and Wrath data over the internet, so hopefully that is not a problem.