Prague + Austria trip

I would go with the 10 days. Even if things are closed on the 25th you still will not be jet lag if you were to arrive on the 26th. If you land on the 26th you will spend 1/2 day going from the airport to your hotel if you are flying into Prague. Hopefully others can give you tips on what to do on Christmas day. I would recommend spending 2 nights in Cesky Krumlov but I don't know what the weather is like that time of the year.

I agree; go with the ten days. Obviously you should double-check this yourself, but from a quick search it looks like the Jewish Quarter synagogues will all be open on the 24th and 25th, although some might close early on the 24th. The Mucha Museum and Museum of Communism websites both say they are open every day, although I thought I saw something about the Museum of Communism being closed on the 24th. Either way, it appears that some sites will be open. And even if they aren't - you're in Prague! You can use those times to wander the city and get over your jet lag. Even if you aren't religious, it might be interesting to go to a Christmas service.

I haven't been to Cesky Krumlov, but I did really enjoy Melk and Hallstatt.

I say arrive on the 26th. The 24th is the main holiday here plus it's a Monday and attractions are notorious for not updating their websites to show closures. Less in the way of available taxis/transport and they'll be charging at Bank Holiday rates. Skeleton Staff all over the place. But if you do arrive on the 24th make sure you leave a nice tip for people that have to work.

If I were comfortable with the idea of walking around outside all day at that time of year in central Europe (which I certainly am not), I'd definitely go for the ten-day trip. Prague is gorgeous. You could spend a week, or longer, just enjoying the exterior architecture.