No, the engine is not dependent on the transmission or Transmission Fluid in order to start & run.
It will start to slip
Stripped transmission, low transmission fluid.
I would say, Low, Second and Drive.
The first gear in a car with a manual transmission is used to start the car moving from being stopped and to climb steep grades at a low speed.
This problem could be the linkage in the transmission. The transmission could also be going out and would need replaced.
For the Reverse problem, you could have low transmission fluid, or a bad transmission. For starting in drive, just about every modern car will not start when the transmission is in gear. They will only start in Park or Neutral. This is a safety feature to prevent the car from taking off into someone or something when started.
Transmission fluid will show full when it is cold, but low when it is hot if it is low on fluid. Transmission fluid has to be checked when the car is warm.
bad fuel, shorted mass air sensor, if the car is automatic might be low transmission oil
speed sensor on transmission
The engine would start and run but it may not want to move.
Low transmission fluid will cause the vehicle not to move. Try filling the fluid and see if it works. good luck
First, I would have your transmission checked. It sounds like the transmission is starting to go. I would rule this out first. Also, you could check the level of your transmission fluid, If it gets too low, it will make it start to slip, if it is low, check for a leak! Hope this is helpful!