Export certificates for artwork
Note that exporting an artwork requires a certificate. If you want to get that yourself you'll have to visit the Artists' Union (19 Xaqani St) on a Wednesday. If the work is over 25 years old and liable to be considered a 'work of cultural importance', you'll need to take the painting to the National Gallery to be assessed. Tuesdays only!
Even new carpets need an export certificate. Start the application procedure at least two weeks before you leave by visiting the Carpet Museum. Theoretically you could go on your own to apply at 19 Boyuk Gala in Old City (entrance around the corner, red bell) but it opens only once a week (currently Thursday at 10am, but previously Monday and before that Fridays just to keep you guessing). Some dealers will do the paperwork and shipping for you.
The old city in Baku has a handful of dealerships which cater to visiting collectors. To get a better deal at such places it may well be worth employing the help of English-speaking guides from Baku's Carpet Museum. They can also help with the certification procedure.