From Mike Miller Tech Talk:
The document I sent answers all your questions, as does the owner's manual in the glove box. Please read both.
Let's go over this again in answer to your specific questions:
1) Gearbox oil drain interval is every 30,000 miles. Time is not a factor. Has the car traveled 30,000 miles since the gearbox was installed (and hopefully drained and filled)? I use and recommend Red Line MTL. Reseal any leaks before filling the gearbox.
2) Differential oil drain interval is every 30,000 miles. Time is not a factor. Has the car traveled 30,000 miles since the last gearbox oil change? If the answer is that you don't know, then change the differential oil now. I use and recommend Red Line 75W-90, which is formulated for limited slip and open differentials. Two new seal rings are required. Reseal any leaks before filling the differential.
3) Brake fluid change interval is once a year, using a DOT 4 brake fluid. Mileage is not a factor. Racing is a factor. This means changing the clutch fluid too. I prefer to do it in spring time, but if the car hasn't had a brake fluid change in 10 years I recommend doing it ASAP. It will probably take two liters of fresh brake fluid to fully flush out the system due to dirt and water. I use and recommend ATE SL or Pentosin Super brake fluid.
DO NOT USE RACING BRAKE FLUID IN THE STREET CAR.
4) I would not use synthetic oil in an engine that has a cork drain pan gasket, like this one.
I use and recommend Kendall GT-1 20W-50 changed at 5,000-mile intervals. Reseal any leaks before filling the engine. At the first oil change, have oil analysis performed including the TBN number by
www.blackstone-labs.com and be sure to tell them you want to confirm the drain interval. I would not worry about the passage of time within reasonable limits. Every single time I have drained oil based upon the passage of time and not mileage, the analysis has come back saying the oil was fine.
5) Engine coolant change interval is every two years. In the case of coolant, mileage is not a factor. Heat cycles are a factor, but coolant analysis is expensive and complicated. Coolant changes are simple on this engine. I use and recommend original BMW anti-freeze mixed 50-50 with distilled water. Drain the radiator and drain the engine block by removing the engine block coolant drain plug. A new seal ring will be required.