I'd say go with a road bike, that's what I use. Fixed gears are a bit of a fad right now but they don't have too much practical advantages besides, arguable, aesthetics (they're one gear, you use your pedals to stop, i.e. no hand brakes unless you add a front one, and are just kind of generally clumsy and can be a bit dangerous to ride, particularly if you aren't already an avid rider). In a fixed gear bike the pedals are tied to the wheel in such a way that you can't coast, if you wanna go down a hill expect your legs to be spinning very fast. If you are just biking around town/park (i'm assuming it's a concrete or smooth path in the park so as to not need a mt bike) and not going up hills you could opt for a single speed (non-fixed but with a freewheel and regular breaks and such) but unless you bike in a flat area (the Netherlands? ) you prolly want gears.
I mountain bike as well and spent a bunch of dough on my mt bike but for my roadbike (which I use for commuting and exercise as well) I hit up craigslist and got a used Fuji (a great company) in great condition for $140 (granted it's 30 years old but it didn't have an ounce of rust on it). Just make sure you find out what size frame you need, you'll want to go to a bike shop to determine that though, then go and look online or elsewhere and find your bike. You can prolly find some sort of approximate conversion online for height/frame size but height isn't a perfect way to determine frame size b/c leg length matters, but for a simple, imperfect point of reference I'm 5'10 and my road frame is 58cm.
As a road biker and mt biker I'm partial to gears and modern braking systems, but this is a fixie thread after all so I'm sure someone can explain their appeal (I for one don't understand it, but to each his own). Plus as fixed gears seem to be the hipster thing to do right now, regular road bikes might be a bit cheaper if you're looking for a used bike.
I used to think pretty much exactly the same as you (read the first page of this post), I live in Hong Kong and we have the worst traffic, hills, pollution and I used to think it would be impossible to get around without gears, but the truth is it's a lot easier on a fixed gear (I have compared it with BMX, MTB and a road bike) so until you have actually tried it, I don't think you should assume..
As for the appeal, to me it's kind of like skateboarding or snowboarding when you are carving down a hill, no brakes and side to side body movement/skidding trying to avoid traffic and pedestrians is a rush in itself, ability to do tricks which are evolving dramatically from other types of riders getting into this sport in the pass few years... Bottom line, you just got to try it! PS I think your 58 frame is a tad too big for you if you're 5'10