I'm no expert, but I'd install the hood first. Any tiling pro should be able to handle trimming tiles to fit. I wouldn't worry about the tilework being specific to that hood shape unless you think you'll tire of the hood before the tile, which seems unlikely. You can try out the hood and make sure it works to your liking before you tile around it.

When we were planning to do a tile backsplash, the tile guy wanted to do the tile before the hood went up.

I'm thinking tile first. I just went out to the barn to look at the hood again and I can't imagine tiling up to the glass edge of the hood and having it look good. The hood will have to be mounted in front of the tile. Also, we're using glass tiles this time and I don't know that they trim as easily as standard ceramic tiles.

Anyone with DIY experience with glass tiles and trimming/cutting them?

If you're going with a chimney hood then you definitely want to tile first. If you're going with an under cabinet wall maounted hood then you can mount the hood first and then just tile up to the bottom edge of the hood.

Tile should come first...just in case you heed to change to a different hood in the future.

Similar question with a bit of a different scenario. Our ceiling is vaulted. We are tiling the backsplash up to the top of the cabinets, not clear to the ceiling. With a wall mounted system, how do we adjust for the difference between where the tile stops and the wall begins so that the chimney will remain flush?

A better way, how did you do this? We also have the same scenario with a vaulted ceiling with tiles finishing half way up the rangehood. (we haven't tiled yet)