1) I understand from the Kiso Valley bus timetable on this website (and having googled for a few hours) that there are no buses running between Tsumago and Magome on a weekday in December. Is this correct? If anyone know otherwise, can you please share the timetable?

2) If the above is true, it leads to the next question: Does it mean the only mode of transport between the towns is by taxi apart from the Nakasendo hike? If yes, how much does a taxi cost and how long does it take?

3) I am arriving Tsumago around 7pm and would like to leave around 6pm from Magome the following day. Is it easy to get a taxi from
a) Nagiso to Tsumago at night
b) from Magome to Nakatsugawa in the evening?

Yes. Unless they change their timetable this winter (which I doubt), there will be no buses on weekdays in December.

If the above is true, it leads to the next question: Does it mean the only mode of transport between the towns is by taxi apart from the Nakasendo hike?

No, you can also travel by train and bus via Nagiso and Nakatsugawa Stations.

If yes, how much does a taxi cost and how long does it take?

From central Tsumago to central Magome it takes about 20 minutes and costs around 3000-4000 yen.

I am arriving Tsumago around 7pm and would like to leave around 6pm from Magome the following day. Is it easy to get a taxi from
a) Nagiso to Tsumago at night
b) from Magome to Nakatsugawa in the evening?

I plan to catch the 4:07pm train to Yokohama but I notice that the bus arrive at 3:55pm. That's cutting it a bit close and I wonder if I will be able to make it with a 12min gap.
1) Do you know if the buses fr Magome are pretty on time in Dec (weekday)?
2) Is the bus station and train station in Nakatsugawa close to each other, and are there many platforms in Nakatsugawa?
3) I hold a JR pass, can I just hop on to the Ltd Exp Shinano or do I need buffer time to buy a ticket?

Last question, can I book the accomodation in Tsumago a day before or much earlier?

Do you know if the buses fr Magome are pretty on time in Dec (weekday)?

Yes, they are usually on time.

Is the bus station and train station in Nakatsugawa close to each other, and are there many platforms in Nakatsugawa?

Yes, the bus stop is just in front of the station and no, Nakatsugawa is a simple train station with only three parallel tracks.

I hold a JR pass, can I just hop on to the Ltd Exp Shinano or do I need buffer time to buy a ticket?

You can just hop on and hope to find a free seat in the unreserved car. Or you could make a seat reservation for the train in advance or just before boarding (the 12 minute transfer time should be enough for making a seat reservation if there is no line).

Last question, can I book the accomodation in Tsumago a day before or much earlier?