Why the obsession with a car? This is not California, where you need a car to get almost anywhere. Germany has the most extensive rail network in Europe, with over 1000 stations in Bavaria alone. Make use of public transportation.
The fastest way (2h17m) to get to Salzburg from the Munich airport is to take the S8 S-Bahn to Munich Ostbahnhof and an hourly regional express (Meridian) from there to Salzburg Hbf. It would take almost as long (2h5m per ViaMichelin) to drive, but you would have to take time to pick up the car. The cost of a Bayern-Ticket for two would be 28€ (your 8 yo travels free with you on the Bayern-Ticket). ViaMichelin estimates the cost for gas alone at 24,15€, plus you might have to buy a Vignette for Austria.
Once you get to Salzburg, there are buses everywhere, but it is a small town, you can walk to most things. There are multiple ways to get to Berchtesgaden from Salzburg, the train via Freilassing or the Watzmann Express bus, RVO 840, from in front of the Salzburg Hbf to Berchtesgaden bus station, and buses in Berchtesgaden go everywhere. A Berchtesgadener Land, Tagesticket-Bus-und-Bahn ticket (12€ per adult, 8 yo free) covers the train through Freilassing or the RVO 840 bus as well as almost all of the buses in Berchtesgaden (except the 849 bus from Obersalzberg up to the Eagle's Nest).
When you come back to Munich on the regional train, you get off at the Ostbahnhof. There is a streetcar from in front of the Ostbahnhof that goes down Berg-am-Laim Strasse (what's the address?). Transportation in Munich (buses, streetcars, U-Bahn, as well as the S-Bahn) is also covered by the Bayern-Ticket that gets you from Salzburg to Munich Ost.
For transportation within Munich, you want to use a Tageskarte (day pass). They come in Single (for one person) and Gruppen (for up to 5 people) versions; you'll want a Gruppen Tageskarte for your full day in Munich. Everything people normally visit in Munich, except Dachau, is in the inner zone (Innenraum), Zone 1. An Innenraum Gruppen Tageskarte costs 11,70€. It gives you unlimited travel on all the public conveyances in Zone 1. If you want to go to Dachau, use a München XXL Gruppen Tageskarte, for 14,80€. It gives you unlimited travel in the inner two zones. On the day you go to the airport, use a Gesamtnetz (entire network) Gruppen Tageskarte, for 22,30€.