Every time I try to tell a suspect to enter the vehicle ( by holding E ), it does the animation, shows them getting in the car, then they teleport out and I get stuck in an animation inside the car. Any ideas why or how to fix this?

Putting an arrested suspect into a car is flawed.
The way around this is to set the suspect down on ground (Q), and then use B and browse to suspect-transport|Local-unit.
Then your suspect will be picked up by an AI-transport
-If you code you can use:
arrestee.WarpIntoVehicle(EmergencyVehicle, 1);

Where arrestee is a defined Ped.
That will allow you to toss a ped into your car
Afaik that is pt the only option.
Using suspect-transport|Local-unit both looks better, and is efficient ...most if the time
Fails happens if the suspect is in conflicting arrest-sequence between you and an AI-cop, You will never win that 'battle' so the best thing is to stand down, and let the AI do everything.
Fail2 is when a suspect cant be accessed through 'E', the your only option is to call persuit-backup, and walk away. There are no fix yet, for these issues.