Your schedule is a little tight, as has been mentioned, I think the thing that throws you the most is the Budapest-Venice-Paris leg...that's a heck of a long way by train, and even if you're flying, you have to count on spending several hours in the airports before the flight, and long transit times between airports and the city, so it still winds up taking the better part of one day getting from one city to the next.
Also, having spent a short time in Venice in January, a few years ago, I would argue that it is probably not the best time to visit. I'm all for checking places out in the off-season and what not, but being right on the water (and under the water, half the time) I found Venice windy and bitterly cold when it was only a couple degrees below freezing - and that was having come from a Lithuanian winter where temps were frequently below -15 C. Because they're right on the water, that damp air cuts right through you. And since Venice is all about strolling and being on the canals, you are constantly exposed to that biting wind.
If you dropped Venice from your itinerary, you'd still have a long haul back from Buda to Paris, but could maybe arrange for an overnight train - or fly. That would give you two extra days to play with, perhaps adding them into your Salzburg and Vienna time - both of which are worth a couple of days, and especially in winter will be stunning. If you wanted to break up the long journey, you could stop in Salzburg en route to Buda, and then Vienna on the way back to Paris.
My general advice for keeping warm is: invest in good, lightweight but warm thermal underwear (silk is good, there are also some cheaper synthetics that work well, ask at a camping specialty store), fleece sweater (warm, light, compact and quick drying), water and windproof jacket or shell, waterproof shoes, warm socks, hat, gloves, scarf. You don't have to worry too much about these items being bulky in your pack, since you'll be wearing them every day, but items that dry quickly are a good idea.
Winter is absolutely magical in northern and eastern Europe - the days are very short, but usually the towns are all lit up with street lights and people put lights in the windows too, so it's a really cosy feeling. You'll miss most of the Christmas markets, but there may still be a few around - be sure to indulge in a drink of gluwein or mulled fruit juice if you find yourself at such a market.