Can you paint over nicotine covered walls?

You will have to wash the walls regardless before applying paint if you want the paint to adhere properly. A simple solution of vinegar and water in a spray bottle or even just mild soap and water, quick wipe down and then paint when it's dry. Not a huge problem.

Ask the clerk at the paint store how to handle that job.

You can paint over them with no problem.

Sure but why not clean the walls first? If the smell won't bother you then paint away otherwise.

Yes.