You have 16 days. London, Paris, and Rome are three of the greatest cities in the world (in my opinion). If you did just those cities, flying between them with budget airlines, I think you’d have a great trip. Each of these has so much to see and experience that you wouldn’t possibly run out of things to do.

You could fly EasyJet between London and Paris. I think EasyJet is a good choice between Paris and Rome, but I don’t remember. Use these sites to find good airline choices:
http://www.whichbudg…
http://www.skyscanne…

However, I can see the merit of throwing in a midsize city or rural area in addition to three major cities. Or you could do daytrips from the above three cities.

I can’t see a night train saving you money with those three cities involved and the availability of cheap flights.

For Rome, I’d recommend a convent room rather than a hostel – although I’m sure you can find a good hostel too.

For Paris, you can often find cheap hotel rooms that are about as cheap or cheaper than a hostel, or some hostels have good double rooms.

For other possible stopover points, what are your interests? What would you like to see or do that you couldn’t do in one of the above cities?

That isnt a very big gap to fill.  About 5 days for each city mentioned. 

why does a war vet need to plan your trip?
 
Your trip isn’t for another half a year, in this period you can’t take 5 minutes of your day to do some research yourself?