I've toyed with this idea as well. In the end I gave it up too. I'm not honking my own horn here, just qualifying my statements. I'm the construcction project manager for a small college here in Oregon.
First the practical concerns. The volume of exhaust created at time of ignition and for the period when the unit is "hot" and cooling (that is above the normal operating temperature) will be very high. We must remember that we are not just exhausting hot air or relatively low combustion fumes from natural gas. We are trying to exhaust gas from wood combustion. Much more toxic than what comes from a stove or oven! You will need a very stout fan during these periods in order to stay out of trouble. I'm talking about a duct fan in the 650-2000 cfm range. These are not cheap. Make-up air is easy in a garage. Just leave the door fully open during these higher exhaust periods (common sense). The hood should be as close (low) over the lid as you can make it and still be able to do your cook. After the unit settles down to it's operating temp., you can close your garage door MOST OF THE WAY. It should still be open 6"-12"(big opening, lots of air available even when mostly closed). I would have the garage door fully open any time I went in to check on the cook. A rate-of-rise or infrared smoke alarm would be very prudent as well (also not cheap).
As for the non-practical concerns, I think that your local building official will require very stringent installation conditions in order to do this work legally. This is if he allows it at all. In most jurisdictions in the USA you can not LEGALLY install a range hood before obtaining a building permit. That doesn't mean that someone has not already done a good job of doing this without a permit! Unfortunately, I can't do the same given my job and relationship with the local bulding officials!
Sorry to be such a wet rag.
Cheers