In movies when an actor gets an injection, is a needle really inserted?

It depends on the actor and the needs of the shot. Generally, if the camera is far back enough to show the injection and the actor's face at the same time, it's easy to use a retractable needle or just remove the needle for the final plunge.

If a highly authentic close-up shot is required for some reason, then there's an edit away from the wide shot anyway and it's easy to swap in the arm of a stunt person getting a legit poke with a syringe where the needle has no opening and the liquid goes into the plunger when it's pushed. Alternatively, for the close-up just swap in the arm of a diabetic person who needs to inject insulin daily anyway. Any serious shoot is going to have an EMT or RN on set to do the deed.

As is often the case with "movie magic", don't overthink things. When you control camera framing, editing, lighting, rigging, etc many times a cheap, fast and easy way to get the shot can be figured out.

No. The needle is retractable. The same goes for knives as well.

https://www.youtube.com/watch?v=2uz1YsCwc4k

It's probably fake most of the time, but I wouldn't put it past some crazy method actors (particularly those who are used to needles) and perfectionist directors to go for an injection for an important shot in the right movie.

I turn my head away from the needle when injected. I don't think it's necessary for the camera to be on the needle.

People who work on films recently had this very question for Alec Baldwin and his crew, with regard to guns.