Is it proper to say "Turn right right here"?

Yes, it's completely accurate. In fact, it involves a literary technique that many authors use in writing. I'm trying to remember the name of the rhetorical device but it involves using different meanings of the same word spelled the same way in the same sentence. A famous instance was FDR:

• "We have nothing to fear but fear itself." (It uses the verb fear and the noun fear.)

Even better example:

• Do I turn right right here?
• Right, right right now!

A third example from Vince Lombardi:

• "If you aren't fired with enthusiasm, you will be fired with enthusiasm."

As you can see, there's nothing awkward about it. Many writers go for making matches like these, especially when they so readily make sense. People aren't confused when you say, "Right, right right now," in that context.

The literary scheme you are using when you say, "Do I turn right right here," is called an antanaclasis. I don't think that a way to say something that is so cool they gave it a name is something you need to avoid using.

Consider, turn left right here vs. turn left, I mean right, here.

It is grammatical, natural, and potentially confusing. Homonyms can be ambiguous in certain contexts, requiring a bit of thinking to decode. Something best avoided when someone is concentrating on driving. Or you can take comfort that you were grammatically correct as the car heads into a telephone poll.

It is technically grammatically correct, but it is a poor choice of words given the context.

There is nothing wrong with it, but you should never use the word right when talking about directions, unless you mean the direction itself. It's not technically required, but I strongly suggest it.

Instead of using right as an affirmation, use correct.

"Correct, right here" is better in the context to avoid confusion.

Or in your example, "Turn right here" can be rough. Does it mean turn immediately or turn in the direction right? It is no grammatically incorrect. However, it is contextually confusing, and should be avoided. It should have been rephrased as "Take a right now" or something along those lines.

Again, it is grammatically correct, yet ill advised given the context.