You absolutely need to use SPIII fluid in this transmission. Your mechanic is right on the money in that using 'compatible' or 'universal' fluids will cause the shaking and shuddering you are experiencing as the improper friction properties of the generic fluid causes the torque converter to be unable to properly perform a partial lock up.
The only solution to this is to flush (NOT drain and fill, you must flush) the system out with proper SPIII fluid. There is no such thing as a compatible fluid for this transmission.
You are going to need to order the fluid from a Mitsubishi dealership. Since this is a specialized fluid that no one else uses, there are no aftermarket equivalents and therefor no one besides dealers sell this fluid.
I have had very good luck with www.mentormitsubishi.com sending products over seas for customers. With a fluid like this you may run into problems, but all you can do is try. If no dealership are willing/able to ship the chemicals to you, you are going to need to find someone to purchase the fluid for you and ship it discretely to you.
You need to order part number ACH1ZC1X05 and you will need a case (12 quarts) for a full flush.
As long as the vehicle has not been in this condition for a long time, a single flush will normally correct the problem within 60-100 miles (a noticeable improvement will occur with 10 miles or so). If the vehicle has been driven in this condition for a long time, it is possible to permanently damage the torque converter. If this occurs, the torque converter will need to be replaced... the flush is always step one however as you can not judge the condition of the converter until the proper fluid has been given a fair chance to correct the situation.
Regarding your water on the driver side, it is the AC drain seal where the drain pipe goes through the firewall. The seal has failed, and as the water leaks out of the car, some of it runs back in along the bottom of the drain tube and re-enters the driver side of the car. You can correct this by either installing a rubber drain tube extension (not terribly easy, and worse on AWD models), or if you are unable to successfully do that you will need to pull the dash and AC box and replace the failed seal on the firewall.
Instructions and part numbers for the hose installation are included in the factory bulletin on this problem found in PDF format here