Good advice. I never install the surface-mounted types of vent hoods that were used in your installation for just that reason. Ice/water membrane will normally help protect against ice dam melt water intrusion, but with your hood being that high up it's probably above the membrane. Somewhat unrelated, yet related...a house I built two years ago, the owners called me up in late January with "a huge, huge, problem". They still have a blanket of snow on their roof, it's been there since the first snowfall in December. Meanwhile, all the neighbor's houses have had snow come, then melt off the roof, then more snow, then again another melt off to expose a clean roof. I drove them around to a few other houses in town that I've built, all my roofs are holding snow. Took them a while to comprehend that it means that the heat they're paying for is staying inside the thermal envelope of their house and not going, literally, "through the roof". The good news is you've identified the problem and it'll be easy to rectify. Good on you for identifying the problem.

From what I read you are allowed to go through the roof however keep in mind maximum length is 35 ft from the dryer and if you use a 45 degree elbow it is 2.5 ft less to bring it to 32.5 ft and a 90 degree elbow if 5 ft less which is then at 30 ft cause every off set is concidere a constriction now i wouldn’t personally exhaust out the roof cause you would need the proper vent hood to restrict rain and moisture from going back in the vent which if any moisture goes in can create lint gathered by the moisture then when the lint that you thought was getting out was collected by moisture then the dry lint that’s been trapped in the vent can cause a fire and also on another note if you exhaust out the roof regardless of the vent being waterproofed you can still get condensation mostly in the fall and winter

Begin in the attic space. Use a string & plumb bob to extend a vertical line from the center of the pipe to the underside of the plywood roof decking and mark the spot. Drill a hole through the roof decking from the underside up through the roof then push a wood dowel, piece of wire or other object through that hole that you will be able to see when on the roof. That will mark the center of hole through the roof. Now go on the roof and locate the center index marker you pushed up. For a 2" pipe I would layout a 3" hole centered on that mark, then cut the hole out with a jig saw. Drop your PVC pipe down through the hole and connect it to the cat iron pipe. If the cast iron pipe has a hub you can get a "Fernco" rubber donut gasket to seal the PVC to the hub. If the cast iron pipe is cut off so that it has a spigot end (raw pipe diameter) you can use a "Fernco" PVC x CI rubber coupling to connect the pipes. Next you will install a "Roof Flashing", which is a flat aluminum plate about 1/8" thick and about 2' x 2' square. The flashing has a hole through the center to fit over the pipe and it has a flexible rubber gasket that seals it weather tight against the pipe wall. The flashing is slid down the pipe and placed tight against the roof. The shingles on the upper side of the pipe should be carefully folded up to allow slipping the upper section of the flashing under the shingles. In this manner any rain water coming down the shingle will go on the top of the flashing plate and continue running down over the flashing and onto the roofing below. The bottom corners of the flashing should be nailed in place to prevent the flashing from moving. Now looking at the pipe from the side you will note the roof angle. The vent must extend 6" above the roof measured from the high side of the pipe on the roof angle. If you live in a region subject to snow the vent must extend 6" higher than your average snowfall. (Check your local AHJ because most communities have set a specific length to allow for snow in your community)

You didn't cite the specific model of question. However, sounds like hubby didn't properly follow the installation instructions. OTR microwaves are capable of external/outdoor venting or indoor/recirculation venting ... assuming there's not something about yours that doesn't support external venting. They are typically factory-configured for recirculation. The configuration is adjusted accordingly for external venting during installation prep, which typically involves changing the orientation of the blower assembly. Hubby needs to review the installation instructions, pull the unit down, adjust the configuration, and remount it.