Thereof, can you paint over cigarette smoke?
Applying paint directly over top of the remaining stains or odor will not cover them up. You'll need to apply a good, solvent-based stain-blocking primer to prevent them from bleeding through the paint. FYI, BIN Primer is also very effective at blocking water-stains from bleeding through your finish.
In this regard, can you paint without washing walls with nicotine on them?
You will have to clean and prepare the walls properly before you paint, to prevent the smoke and nicotine from bleeding through again. Wash the walls with an aggressive cleaner and then seal them with stain-blocking primer to ensure that stains don't creep through later. Move the furniture away from the walls.
The best choice for covering nicotine stains in a single coat is a white-pigmented shellac primer.