There's no reason you can't install a hood on the brick, it will work fine. You need to buy one designed to hang on the wall, not the ceiling, and there's loads of choices either way. You can run a power line up from behind the range and hide it with a splashback. I highly recommend a splashback instead of having bare brick as it will be much easier to clean food off of.
As for ducting it completely depends on the construction of your home how possible it is to do, and how cosmetically pleasing you can make it. It's not possible to say without more detail.
Before making any decisions you should check your local building codes, some areas mandate you have to install a duct, others don't. Some mandate you can't have your range in certain places. It would stink to build your kitchen and then find out it's illegal!
A classic approach in cabinetry is the use of filler strips. These are pieces of wood, matched to the finish of the cabinets that are used to fill gaps, usually between cabinets. Often the horizontal spacing does not work out to exactly the width of standard cabinets (especially in renovation work). You design to the standard sizes slightly narrower than the overall, and then add a vertical filler strip between, or at the end of a cabinet, to fill the gap.
The same principal can be used to fill a gap between the top of the cabinet and the ceiling that will then give you a horizontal surface to attach the crown molding. A strip will probably have to be tapered to fill much of the gap, but it does not have to come down to a point because the molding will cover much of the gap. The strip can be glued into place,since it is purely cosmetic, but be careful when nailing in the crown.
The biggest question will be finish. If the cabinets are painted, you may be able to buy a filler strip from the cabinet manufacturer and run it horizontally instead of vertically.
For stained cabinets, you can do the same, but the grain will run perpendicular to that of the cabinets. It is a matter of taste as to whether that is acceptable to you.
You also could put in a homemade filler strip (plywood, clear poplar, mdf) and paint it to match either the cabinets or the the crown molding. If yo paint it to match the crown, the effect will be that the crown looks like it is a bit deeper at that point, but probably will not be noticeable.
Best Answer
Cut horizontal areas behind your wall cabinets for new 2X4 or 2x6 nailers set at the tops of the wall cabinets so they are in line with the screw rail of the new cabinets. Cut out the firring strips too. Cut the where the ends of the 2x will be an inch or two behind the edges of the cabinet, so if the wall paint chips or paper tears, it will not show.
When you cut out the wall, the 2X material should be very near the thickness of both the firring strips and drywall, 1/2" drywall plus 3/4" firring strips, should mean you will need to thin down the 2X material a bit. A table saw can do that with care.
To install the 2X, you can use flat head expansion anchors and construction adhesive
Place an anchor on each end and through the rest of the length of the 2X, about 8-12" apart. Place a few large beads of construction adhesive on the back of the 2X after you drill all the holes in the 2X, but before you drill the brick. It is important to get the anchors in the brick, not the joints, although the horizontal joints (bed joints) may work, surly the short vertical joints (head joints) you need to stay away from. If the brick is a "core brick" a term used for brick with holes in the inside, do not place any anchors in the middle of the brick, only near the ends. If you know them to be solid brick, then it will not matter.
After the blocking is in place, secure the cabinets with what should take 2" screws, if your cabinets are made with 1/4" backs and 3/4" screw rails and 1" to go into the new blocking you just added. Ideally the screws could be 2 1/4" but typically are not available in that length. 2 1/2" screws could work, but you are taking the chance that the point would "bottom out" on the brick surface before the screws drew the cabinets tight to the wall. I have had screw points dig into the brick a bit before they started shoving the strip off the wall, but that is the chance you would be taking. IMO, not worth the chance....