I'll be playing on a euro server...are there many players still running around?
Technically, there are no EU-specific servers; FFXIV divides it's servers between the JP servers (which are located in Tokyo) and the NA/EU servers (which I think are located in Montreal, Canada). However, the unofficial server website http://www.arrstatus.com/ lists Ragnarock (Legacy), Cerberus, Lich, Moogle, Odin, Phoenix, and Shiva as "Europe", so make of that what you will. As I play on Hyperion, I can't say what the population of those servers is like, but there ARE server-specific subforums here that you can take a peek into and see how things look.

Is there a lot of grind?
There is going to be a grind in every MMO ever, that's just the nature of the beast, as making people work for a reward is what keeps people playing. This question is otherwise hard to answer, because it depends on what kind of "grinding" you're talking about. Are you talking about grinding EXP to reach the level cap? There is some, but it's not as noticeable as some other games. Even after you run out of quests to do, there's still clearing dungeons, accepting levequests, and participating in FATEs to earn you EXP. For crafting and gathering classes, the EXP from levequests is very helpful for leveling up. Are you talking about a grind to earn gear? There is this, but that's the nature of the RNG. You can stick to crafted gear, which is more readily available, but you can't get the best gear without wrestling with the RNG.

Hows the pvp?
Haven't played it, can't comment.

Game economy?
Entirely depends on the server, but right now things are "bad" because of housing. People were really expecting houses to be more readily affordable, and thus aren't spending money to try and save up for them. The price of houses have already fallen quiet a bit, and when more people feel willing to afford them and spend money, the economy should pick back up. There's also the 2.2 update, which Yoshi-P has said should revitalize the economy with new crafted items.

End game stuff?
Don't ever let someone tell you there's no challenging endgame. How challenging something is varies from person to person, and what might be easy for one is still hard for another. Right now, there's not a whole lot of "endgame" content outside of the Binding Coil of Bahamut and Crystal Tower raids and the Hard and Extreme-mode primal fights. 2.2 will add a new hard and extreme-mode primal fight as well as 4 new turns to the Binding Coil of Bahamut.

Server issues?
None really to speak of, aside from maybe some lag (which isn't always the server's fault).

are there pvp battlegrounds or arenas?
The only PvP right now is arena-based, with 4-8 person teams. There's a new PvP mode called "Frontlines" that has apparently been mentioned in the past, but I have no information on what that is or what form it'll take.

is there a real money shop by the way? I don't intend to spend any money on extra items and such
There is no cash shop. FFXIV earns it's income via player subscriptions, and thus has no reason to make people shell out even more money for things, however trivial they may be.

could there also be a price cut or deal coming soon? I know that 16 euro for a digital version or 24 euro for a boxed version isn't that much for a game like this but would still be nice to know
I don't think SE is going to drop the price of the game yet, as it's still fairly fresh on the market and doesn't even have one expansion out yet (and might not even drop the price after said expansion, either, but you never know). Buying the game digital is about as cheap as you can probably get it.