I will explain the situation as well as possible so that you can imagine the worst amount of damage. I pulled up in the school pick up line and my left bumper was lined up to the right bumper of the car in front of me. He was parked in the middle of the lane. I was parked about 20 ft behind him. I had my foot on the brake and the car still in gear. He apparently didn't see me and floored it and hit me. My bumper folded up into my tire and it pushed my Pacifica diagonally. I put a hole in my bumper and completely shifted the front of the car to the right. Their is a noticeable gap in the hood and the driver door. The whole bumper will need to be replaced. The fender will need to be straightened and painted. When I got home after the accident we looked the car over. I did not notice leaking oil. A couple days later I did start to see puddles of water coming from my car and noticed that when I went to take off it did not want to shift into the next gear. It wants to go up to 3000-4000 rpm's. I quickly let off the gas because I know this is not normal. I also noticed that the middle console inside is shifted to the right. When I got the estimate done, they could see that the A/C Condenser is leaking and the oil pan was wet. I told them about the transmission issue and they said they could not put that into the estimate because they cannot prove that it is apart of the accident until they take everything apart. This issue did not occur before the accident. No other auto shop will touch my car because of the accident. Is it possible that this person hitting me at the elevated speed and angled while I was in gear could have affected my transmission? I will add that the speed was great enough that the other cars frame was bent and their car was totaled.
A good collision like you explained can cause all sorts of problems. The body shop is correct in telling you that he can't tell you the exact problem until the vehicle is taken apart.
Once the vehicle is apart the mechanic (body shop) can inspect the frame, body, electrical harness and wire connections as well as transmission parts like shift cable and the case (body) of the transmission to make sure there is no structural damage to the transmission itself. If the problem is related to the collision then it would have to be shown to the insurance company to have the estimate of the damage to the vehicle recalculated.
Once the vehicle is repaired the body shop should analyze your vehicle with a computer scanner to make sure there are no trouble codes (check engine light) and he can also diagnosis the transmissions electrical operation to determine it is working properly.
The mechanic knows what the problem is but cant submit it to the insurance company until I am out of a car. Should I tell the insurance company that my car is not shifting properly and that there is an oil leak? The shifting is bad enough that another accident could happen. I do not have another vehicle to drive and I have three children. Who is responsible for this problem here?
That is what you tell the insurance company exactly; the car is a safety hazard and is unsafe to drive with not only your children in the car but to yourself and others on the road. The shifting problem is a safety issue and could cause more damage to the transmission and or engine the longer you drive it.
Most insurance companies should pay for a rental or loaner car. This sounds like it would be the other insurance company's responsibility and not yours but ask your company if they would pay for a rental if the other company does not.
The oil leak could cause engine damage if it gets worse if that is where the oil leak is coming from. It would also have to be determined if the leak was caused by the accident.
Try to make arrangements to have the car fixed as soon as possible for the safety of you and your children and also other motorists.
Driving the car longer than you have to could cause more damage where the insurance company won't pay for anything that happens to the car after the original assessment of damage.