Simple answer is maybe. If you are planning on putting in lots of miles in undulating terrain to improve your fitness, then maybe a geared bike might be a better solution, especially if road riding and big distances are new to you. It might be a bit more fun that way. Even after 15 years+ in the saddle, if I'm planning on doing any decent distance (20-30miles+) then I always take the geared bike. Fixed wheel and singlespeed are great about town, for short blasts in the country and for crazy audaxers who enjoy pain. They do force you to get fit though.
Riding your mountain bike in one gear will give you an idea of how a singlespeed would feel, and the tyres will only really effect your speed, not necessarily the experience (hard to say how it would effect the gearing).
Going fixed is a whole different beast however, and IMO if you are going to give one gear a shot, go the whole way and make it fixed. It really is a different experience, and a good one at that. Strangely, it feels a bit easier on hills to a singlespeed with the same gear. A bit more efficient on the way up, but obviously without the ability to rest properly on the way down.
Most off the shelf bikes come with flip/flop hubs, with fixed on one side and singlespeed freewheel on the other, so there's no need to go all the way if you fancy easing yourself into it. Just turn the wheel round to suit the conditions/fitness/mood.
Fixed feels strange at first, and unless you're careful you'll scare yourself a couple of times before getting the hang of it, but it only takes a day or so before it becomes second nature. In fact, after a week or so, going back to a freewheel bike feels odd.
And don't worry about the hills too much, I'm vastly overweight, fairly unfit, and live in hilly Edinburgh and I manage just fine on a fairly high 73" gear. Sure, it's hard work, but what doesn't kill you only makes you stronger.
PS: Welcome to the forum. Where about in the Borders are you? I often head down that way when the weather's nice (well, Peebles and back is about as far as I've got so far

).