I would put the hood over both because my Jenn-Air Grille/Griddle cooktop is horrible at venting. My cabinets always get a ton of condensation on them because of how inefficient the downdraft vent is. I am currently planning a kitchen for our vacation house and I definitely am using a hood over the cooktop because of living with the Jenn-Air.

GE, GE Profile, Jenn-Air, KitchenAid and Whirlpool all make cooktops with integrated downdraft, and are priced from $529 - $1280.

220/240 circuit, separate circuit breaker. There's probably 110/120 there now, for the gas igniter. Check the amperage of the wiring as well as the amperage requirements for the cooktops you are considering; often you will need 40-50 amps, while standard 110 is usually 20-30 amps. 110/120 with its own circuit can usually be switched to 220/240, but if the wire is too small (amperage) then you'll need new wire installed. Also, some cooktops are hard-wired & others use a Hubble rather than a conventional plug--check this out before you have the wiring work done. I highly recommend induction, with the highest wattage you can find for your budget & space. If you are nervous about induction or have great pans that won't work on induction, get a combo unit that has some induction & some electric burners.

Not only are they not "out of style", but they're still as common as they've ever been. Jenn-Air's website alone has about 6 different 30" or 31" electric cooktops with this feature (called downdraft). Kitchenaid is the same company as Jenn-Air, as is Whirlpool. It would be easier to name the brands that don't offer them. In addition to downdraft cooktops, you can get a standard 30" cooktop with a separate telescopic downdraft unit behind it, that raises from the countertop when in use and retracts the rest of the time. Or you can get two 12" to 15" wide cooktops (or grills, wok burners, or fryers) and put a separate downdraft vent between them - these tend to be pricey (i.e. Gaggenau) but offer lots of customization.