2 questions: Time to purchase train tickets/railpasses and locating Army quarters in Berlin

You can typically buy rail tickets 90 days in advance on the DB website; this will give you access to the least expensive rail fares. Very true train schedules can change so be proactive after purchase and keep checking the DB website for changes. DB cancelled our November 6 train from Dortmund to Amsterdam Centraal. Luckily DB emailed us 8 days earlier. They don't always do that.

As for your 2nd question, perhaps email the US Embassy in Berlin?

If you google "Army Barracks Berlin 1963", you will get some possibilities in Wikipedia. The barracks are named and can be further googled. There is a lot of information available and it will take research. What would make it easier is if you know his Unit or the type of service (military police, infantry, etc.).

"...take a train to Munich, later Fussen, then Salzburg and back to Munich."

The only ticket you need to pre-purchase is Berlin - Munich. The others are best done on local day passes. The Bayern Ticket is the most famous of these and covers the final 3 trips - buy one for each day, and buy each at a ticket machine in your city of departure on the day of travel, and you still get the lowest possible price. There are no reservations for regional trains as a rule (these are the trains to be used by Bayern Ticket holders.) If you pre-purchase the Bayern ticket, it is NOT refundable if your plans change:

http://www.munich-touristinfo.de/Bavaria-Ticket.htm

For travel between Munich and MUC airport, you likely need this ticket:
https://www.mvv-muenchen.de/en/tickets-and-fares/tickets-daytickets/airport-city-day-ticket/index.html

Note that the Bayern ticket is one price for all three (or up to five) of you -- write the names on the back of the ticket. Easy to use. If you buy from a manned ticket booth, as opposed to a ticket machine, there's an additional €2 service fee. Also includes local transportation connections in Munich, Füssen, Salzburg on the ticket date.

I use this website when I am looking up information about the military in Germany from 1945 on. You should be able to find the Kaserne or post where your FIL was stationed. Use Google Earth to match the location up with today.
Many of the good tour companies in Berlin will have information about the locations of the various posts as they often have guides who specialize in this kind of information. Try Insider tours.
There probably isn't anyone at the Embassy that would know this.

http://www.usarmygermany.com/Sont.htm?http&&&www.usarmygermany.com/usareur_city_hanau.htm

You can typically buy rail tickets 90 days in advance on the DB website

That changed a while ago; now it's 180 days. Booking popular connections only 90 days out may imply that the cheapest tickets already are sold out.

Re Q2, my neighbor was there in those years and says this: "...about the location of the NCO housing. It is near the US Consulate...not Embassy, on Clayalle. They would also find the Allierten Museum (Allied Museum) which is across the street very interesting. The housing area is behind the Allied Museum, but is now housing for Berliners. Another Landmark and the closest U-Bahn (subway) is Oskar Heleneheim U-Bahn station." Enter "Allied Museum, Berlin" in google maps and it comes right up.

Hope this helps.

As most of your your travel is regional, reservations aren’t even possible as those trains don’t have seat numbers. Buy your Bayern Tickets the day you travel. I googled Euraide in Munich and they are open today. Perhaps the European system of date writing - DD/MM/YY - threw you off as it likely said 3/1/19, which is Jan. 3.

Maybe an old timer at the Checkpoint Charlie museum might be able to help you. Or perhaps a Berlin Wall archive on BernerStrasse might have the information. They have a lot of documents there.

Thank you to everyone who has provided me information! This is very helpful. The more I learn, the more excited I am about making this long-planned for trip!

Cindy

Note that the Bayern ticket is one price for all three (or up to five) of you

Not sure what you mean by "one price", Laura. The Bayern-Ticket.*** used to be*** one price for 2 to 5 people but they changed that a few years ago. This year it is 25€ for the first person and 7€ additional for each person from the 2nd to the 5th. For 3 people (assuming your daughter is over 14), it would be 39€. If she is under 15, you only need a two person Bayern-Ticket for 32€. Children under 15 travel for free with a Bayern-Ticket when their paying parents or grandparents buy a one or two person ticket. The first person has to be a parent of the children; the second person (only one) can be the other parent or any other adult (like another child over 14).

That changed a while ago; now it's 180 days.

So you can now buy tickets for anytime through early July.

You can purchase your Berlin to Munich tickets (with reservations) online from the Bahn now or whenever you feel you can commit to a specific train (date and time). Buy your Bayern-Ticket on the day of travel from a ticket automat. No reservations are possible with the trains usable with a Bayern-Ticket. I see no reason to involve Euraide.