Cambodia could work because Siem Reap is inland and the amazing Angkor Wat temples are worth a full day or two, and the town of Siem Reap itself is quite nice. You could actually combine a couple days in Siem Reap with a few days in Bangkok, and take a (cheap) luxury bus between the two. Bangkok is a really fascinating city with a lot to see, plus it’s a great transport hub. The other main non-beach option in Thailand is the city of Chiang Mai, which is wonderfully inexpensive with quite a bit to see. It’s even a bit cool during December, so that might be a nice change of pace from India. Many people are also going to Myanmar these days, although I haven’t been yet so it’s hard for me to recommend it. I’d say if you combined 2 or even 3 of these places you could have a very interesting holiday.
I think a side trip or two will be a great idea. Singapore is a large and very interesting city, but you can see everything that you want in maybe 3 days or so. You have almost limitless options if you include the possibility of a flight. Even though Singapore itself (especially hotels) is kind of expensive, you can get cheap flights to all points of Asia from there, and the airport is awesome. Air Asia and Tiger are two great low-cost airlines with many flights out of Singapore, so you could check their destinations and find something good.
This is a huge question and I’d need to know more about your preferences and your budget to answer it properly. But I can give you a short version here. In my opinion, which is matched by many other people, the most interesting and dramatic cities in Europe are London, Paris, Amsterdam, Rome, and Venice. Visiting those would require about 16 days to do it properly because you’d want to spend 3 or 4 days in each city, except maybe only 2 days in Venice because it’s quite small.

If you’re lucky enough to have a Farmer’s Market in the city you’re visiting – give it a try! They usually have vendors there that really take pride in what they’re making. You can get some beautiful farm-to-table meat and freshly picked vegetables right from the farmer themselves. It makes for a great experience, and gives you some fantastic product to eat.
Traveling during the last ten days of December means crowds and exorbitant prices throughout much of the world, but the first two-thirds of the month—before holiday airfares and hotel rates go into effect—can be a serene, value-laden, and just plain lovely time to vacation. Europe is all decked out for the holidays—with locals generally friendly and in high spirits; in the Southern Hemisphere, spring is in bloom; and in the Caribbean, Hawaii, and Mexico, hurricane season is over, the weather is gorgeous, and there are free upgrades galore.