There are basically three types of equipment venting: gravity venting (or natural draft), direct venting, and power venting (or mechanical draft). For gas fired water heaters I normally just see gravity venting (with the open type draft hood at the top of the water heater), and for boilers/furnaces either gravity venting or power venting. If I came across direct venting I would probably check with the WH manufacturer to make sure thats right for the equipment just because it’s something I wouldn’t usually see.
Same basic checks as a typical gravity vent WH, except for a few things like checking that there are tight connections on the vent (unlike a gravity vent which is intentionally open at the unit) and that the side-wall vent hood is at least 12" above grade (more for areas with significant snow).
Sounds unusual that the WH wouldn’t fire with hot water in the tank when you turned up the temperature. Was it close to the maximum temperature setting already, and did ya wait for a while to make sure it wasn’t just a time delay.