Oh please help me with our itinerary!

Since your 9 days will all be in Bavaria, you will want to use a "Bayern ticket" on the days you are travelling outside Munich.
This is a one day ticket that is valid on all local trains (not IC or ICE trains, including U-Bahn and trams) and most buses in the state of Bavaria (plus as far as Salzburg).
You buy it at any station ticket machine. Price €25 for 1 person, €32 for 2 people, €39 for 3.

Details here: https://www.bahn.com/en/view/offers/regional/regional-day-ticket-for-bavaria.shtml

Map showing the rail lines it is valid on here: https://www.bahn.com/en/view/mdb/pv/deutschland_erleben/allgemein/tickets/pdfs_laender-tickets/2015/mdb_190311_streckenkarte_bay_2015.pdf

Munich is a great large European city, and there are plenty of things to do. I like to think of it as a 4-5 day city.
Salzburg is 2 hours to the east, and it is relatively small in size-- more of a 2-3 day city.. You could make a day trip from there down to Berchesgarden or even over to Hallstadt.
Instead of the trip over to Fuessen (which we have made), have you considered taking the train from Salzburg down to Innsbruck to see The Alps. The scenery is absolutely breathtaking from there. It's just a 2 hour train ride back into Munich as you're flying out.

One of my ways to find rooms is to go into Google Maps and enter "Munich Hauptbahnhof hotels" where the following pops up: https://www.google.com/maps/search/Munich+Hauptbahnhof+(Main+Station),+hotels/@48.1413048,11.5568239,17z/data=!3m1!4b1
A Google map will have blue blocks that have a price of the hotel per night. You can click on the block and read all the rental information in the left column. Also enter your expected dates you'll be in town. It's a great way to find accommodations that are close to the train station--for walking purposes.. I never spend even $200 on any room in any city anywhere.
We tend to roll with the flow on our trips, and I often buy rail tickets in the station as I hit town--unless I'm traveling on holidays. When possible, I may buy tickets ahead if I'm sure of my dates to get discounts, but never the day I arrive in town by airplane. See above for the Bavaria discount pass.

Hi Cleo= This one is easy.
Stay in Munich first 2-3 nights in hotel near HBF- there are several right across the street in your price range. Do day trip by train to Fussen . It is about 2 hours each way so your total train time is 4 hours.

You might want to overnight in Fussen and catch the train to Salzburg the next day. This is a train back to Munich then Munich to Salzburg, about 2 hours each leg or 4 hours total train time.

If you do not spend a night in Fussen, catch the train from Munich to Salzburg. 3-4 nights. Day trip from Salzburg to Berchtesgaden by bus. Pick a good day (No rain) for Berchtesgaden
(lake Königssee ). Eagles Nest is optional unless you want to make a special trip to see the bunkers and document center - the top isn't much but a restaurant and some scenic views. There is plenty to do in Salzburg for the rest of the time. Halstatt is a nice 1 day daytrip by train if you get bored in Salzburg.

Train from Salzburg back to Munich to spend the rest of your visit. Train time 2 hours.

Sorry, you won't be able to do anything at the Kehlsteinhaus - often called Eagles Nest in English - because you will be there 2 months before it opens for the season. https://www.kehlsteinhaus.de/english/

Are you talking in US or Canadian dollars - or Australian?

I recommend that you visit the Romantic Road that runs from Wurzburg to Garmisch/Fussen. Several interesting walled cities right out of the Middle Ages. Rothenberg on the Tauber is the most notable. Dinkelsbuhel is similar and without all the tourists.
You can find a tour or rent a car and drive it yourself. Traffic is not heavy in the countryside.