There are a number of events and activities in the game that will not be available in 2.0 by simple virtue of the things changing (for example: current achievements for unlocking all the aetherite gates in a given region will no longer be obtainable because all the maps are changing). There's a full list of which achievements will no longer be available (from which you can draw conclusions about which activities will not be available).

As far as player data? All of it--so far as we know--will transfer. That does not mean that it will all have the same value. Given that abilities are being reworked, many classes may play slightly different. Additionally, because numerous gameplay mechanics will be changing, you can expect the value of certain things carrying over to change. Whether this shift in value will be negligible or dramatic is very much up in the air. At this point, though, I'm refraining from spending any great deal of gil in game on any gear since there's no way to know how much stats will change.

Given how so many of the stats currently make almost no god damned sense or have very little appreciable value (things like crit rate +1 or enmity +1 only actually represent a 0.1% observable increase), I expect that a lot of the math going on in the background could change. So while your awesome triple melded belt might totally be worth the few million for the next 50-odd days until they take down the servers, you might wish you had that gil back in your pocket when 2.0 kicks off. Irrelevant for those who haven't joined yet, but it's something to consider for people who are already at endgame. That said, your triple melded belt could also suddenly be worth significantly more. I expect a great deal of re-balancing under the hood to accomodate for a lot of the changes they're rolling out.