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You've probably got a leak in the tire. It is most likely a very small hole and a slow leak, which explains why it doesn't just go flat. Sometimes the place where you bought the tires will fix this for free. The tire probably just needs to be patched.
This can be pretty tough. When you search in the directory for shops that work on Toyotas and have ASE certified technicians, you get these three: http://repairpal.com/directory/289007-castle+toyota http://repairpal.com/directory/290733-5+star+auto+clinic http://repairpal.com/directory/2186-kazimierz+lipinski ASE certification is helpful for picking between shops. You can also try search your area and some of the surrounding areas on www.iatn.net, which can be useful for finding good technicians. Good luck!
I don't know this car specifically, but there's probably a speedometer cable and a sensor. The problem could be at either end--with the sensor or with the connection at the dash. I recommend getting a repair manual for this car and checking the diagrams--they probably have a manual at your local library. That'll tell you whether or not you can replace the part yourself--could be a big job.
Do you find fluid on the ground after parking for a longer period of time? It could be a simple leak, which may be easy to fix; however, the transmission may be damaged and need repair, which can be very expensive. The best thing to do is get a good diagnosis from a technician you can trust--they can help you find the most cost-effective solution.
I recommend trying to get your hands on a repair manual--you may be able to check one out at your local library. Interior disassembly can be a pain and is complicated, and you'll really save some time and trouble by having access to a diagram that shows you where the fasteners are.
Unfortunately, cracked cylinder heads can be difficult to fix, and they are always expensive. the best thing would be to replace the cylinder head, but if the part isn't available, then that's a problem. It may be possible to fix the crack and keep the car drivable. I recommend calling some other shops and even parts stores to see if anyone can offer to a less expensive solution than replacing the entire motor. Sorry you're having such a mountain of trouble!
Going with the owner's manual is the best thing that you can do. The manufacturer has recommendations that are specific to your car, so you can ask the dealer to include what the manufacturer recommends and ask for them to justify everything above and beyond that.
That shudder associated with anti-lock brakes should only occur under circumstances when you are braking very hard and the wheels are starting to lock up. Wet conditions are a great example of when it happens more frequently. It is unlikely that the ABS system is coming into play when you are parking, which is probably at low speeds. Since you say that it happens only when the wheels are turned, it could be a problem with a steering component or a suspension component, and not be related to the brakes at all. If it happens every time, it should be easy for a good technician to diagnose and fix.
BMW 325i's are very reliable cars when they are well-cared for. The information I'm looking at says that BMWs made in the early 90s require what BMW calls and "Inspection I" at least every 10,000 Miles, according to factory specifications. This service includes changing the oil and oil filter, and possibly the air filter. What service you should perform now also depends on what was last performed by the prior owner. The best bet for keeping the car tip-top is to take it to a BMW expert and have them service your car based on what was most recently done and what it appears to need. The oil leak on the engine is definitely a concern. I wouldn't recommend mineral water--distilled water is best, or at least clean water with low mineral content in an emergency, but maintaining the right coolant mix is very important. BMWs are sensitive, high-performance cars, and investing in maintenance will save big bucks in the long haul.
The multimedia systems in these newest cars are hugely complex, so I don't know of any solution for you. Maybe try a Mercedes enthusiast site? There may be some seriously knowledgeable hackers out there. One thing to look forward to? I read recently that Mercedes is working on "split-screens" that will be factory equipment in there cars and show the driver the navigation screen and the passenger a DVD.