You will get a lot of mixed ideas on this, first of all remember if you paint it will take away from the original look like alot of folks like. Also it will take away from the re-sale value if you do decide to sell in the future. I will tell you this I repainted my bathroom a retro green and added vintage florida postcards all over the walls I'm still looking for just the right material to make shower curtain and window curtain I just love it but I know it will hurt the re-sale value but I didn't care because I know the Airstream will never be sold. I'm still working on how I'm going to carry it with me when I die.
I would think hard about painting the fake woodwork.