Advise needed for Travelling to Eastern Europe in December

well I wouldn't call these places eastern Europe, central at best.what dates are you travelling as any advice would have to be based on whether you are visiting any of these places over Christmas.
I have to ask why are you using Munich to enter and leave from when you are spending no time there?

Looks nice. My one comment:

Day 12 - overnite train to Prague

This train leaves you In Prague at 4 am. What will you do for the next 2 hours before the city wakes up? Better would be take the 7:25 EC train in the morning which will get you to Prague in the early afternoon in time to check into your hotel, freshen up and have an evening on the town.

Flying into Munich after a long, long flight will be physically and mentally tough.. It's one of our favorite European cities with great tourist sights worthy of a 4 day visit.
A couple of days in Salzburg would be adequate as it's just not that large of a town. You could also use more time in Vienna, another very important and great city.
Budapest and Prague are cities that are very much alike. I would suggest you skip Prague.
Getting back into Munich by train for your outgoing flight is also a little easier from Budapest than Prague.
Travel for 7 adults and 3 children could get complicated. I suggest using a good travel agent for so many people versus making all arrangements online.

Book a multi city airfare - in Munich, out Vienna.

Day 1 Arrive Munich
Day 4 To Salzburg
Day 7 To Budapest
Day 10 To Prague
Day 13 To Vienna
Day 16 Depart Vienna

Why visit in December? Is it because of the Christmas Markets?

We lived in Augsburg, Germany for four years and loved the Christmas Markets, but Christmas time it gets dark at 4:30pm. Also, it will be cold. You might get snow and driving could be a problem unless you have chains for your tires.

Perhaps you plan to take the train, that might be wise.

If you do this, bring a nice warm overcoat, gloves and a warm hat.

Another option is to take a river cruise. Several lines have what the call the Christmas Market cruise that goes to the places you mentioned.

Keep in mind that the night train from Budapest to Prague is direct but arrives prior to 0500, if that matters to you. I myself prefer arriving ca 0600. Prague central station is big, modern, bright, spacious, with all the latest tech as you would find in any big modern central station in Austria and Germany.

I like itinerary suggested by djp_syd. To avoid backtracking I would slightly adjust it:
Munich
Salzburg
Budapest
Vienna
Prague
depart Prague
For Salzburg two days, all others three days.

I'd take a day from Salzburg and a day from Budapest and add to Vienna.

Thank for all the advises.

My thought above re departing from Vienna was predicated on my thinking there are direct flights with SQ from Vienna to Singapore. Unfortunately there are not, so you are limited to the Munich option. I would go with 3 night stays everywhere.

Day 1 Arrive Munich, then to Salzburg
Day 4 To Vienna
Day 7 To Budapest
Day 10 To Prague
Day 13 To Munich
Day 16 Depart Munich

All I can add is that Budapest requires at least 3 full days (4 nights) and Czech air has some pretty cheap flights non stop to Prague from Budapest.