What am I left with after rooting? I have a stock Nexus 5 and I want to root the device so I can tether it with a laptop.  I see instructions on doing this, and it all looks plenty easy, I am just concerned with what's going on and the mention of images.

In the end, I just want to be left with my same stock android, just with the ability to do some things with elevated privileges.

After the root process is the phone executing any other code or was that code just a temporary thing as part of the process to root the device?

Also, if/when a new Android OS update comes along, will I still receive it like I normally would?
After rooting you will still be left with stock android just with elevated privileges as you've said.

Remember, your data will be wiped, so I'd recommend doing a backup first!

If a new OS update comes along you will not be able to install via the normal Over the Air method, but you can still download the update manually and flash it via TWRP.

If you take a look here you can see all the other things you gain by rooting.