Regarding interchangeable components, it's hard to say without pictures, but it's likely this will be hard because a) old bike, old standards and b) even with the same standards road and mountain components are substantially different.
Regarding old vs new, this question is hard to answer. If you take a look at similar answers in this Stack Overflow you will see big discussions regarding old vs new bikes.
Some people say they have old bikes that use everyday reliably, and that's great. They also say performance is really nice, and that is great either. Other people love their old bikes. This is all great, as what is really important is that you have a bike and that you have fun in it.
However, I have a different take, but given what's stated above please keep in mind this is (a little) a personal opinion, and that people will disagree for the reason above, some time rightfully, some times not really. I am also the owner of old and new bikes and deal with bikes daily because most of my groups of friends have bikes and we all use them daily.
An old bike is an old bike. It's like an old car. I have a 15 years old Toyota that works reliably, absolutely nothing is wrong with that car. I love it. Does it perform as good as my parent's 3 year old one? No way. Does it require more maintenance? Yes it does. For two reasons: time takes an impact in every vehicle, everything has a life span; and technology evolves and changes in 15 years.
It is the same thing with bikes: you can have an old bike that works well, and you can use it everyday, but a new bike is a new bike. Having a component that just left the factory is just better than using one that that has been used for the last 10 years. Worst case scenario, and your new bike even comes with guarantee, some of them of a few years. Not only it is more likely you will have problems with an old bike, on the short term that is something you will have to deal with it, instead of simply filling a guarantee claim and get a new one.
Some people will now come and say that "mechanic principles didn't change that much in the last 15 or 20 years". That is true. A derailleur works the same way. And so do the brakes. But returning to the car example, an engine has the exact same 4-step gas-consuming principle as it had 20 years ago, and driving a new car is much, much better. It's faster, more powerful, revs easier and consumes a lot less fuel. And it is the same with bike components: new brakes have better materials, tweaked designs, servo technologies, and therefore they break much better. This is true for all bike components, as they didn't suffer a complete revolution but they sure have suffered years of evolution.
So if you felt in love with old bike, want to go through the process of restoring it or buy it ready to ride, maintain it, and have fun riding it, that is great. Again, having fun doing it is what really matters, and if your friends are not Cipollini's I am sure you will be able to catch up with them. However, I would recommend just get something new.
I have old and new bikes, but for long rides I always use the new ones. I love the old ones much more but really, being on a modern bike with modern components is just different. It is just better.
You can get something pretty good close to the $1000 range, as you can see here. For example, I am not sure you can get find B'Twin bikes in the US, but one of the bikes on the list is a B'Twin Triban Black 5 that goes for around $700 and has five stars.
Another thing you can try is to find shops around you that rent bikes. Rent one for one or two days, and get the feel of what riding a road bike is. It is very different from mountain biking, but I think you will love. I do, but again I love every kind of bike ride.