Is the open grate in the ceiling between the garage and attic necessary

If the furnace and water heater are gas powered, it would be to provide adequate airflow for their combustion so that you don't get carbon monoxide forming in the utility room.

Here's a link with some information. An excerpt:

The heating appliances are fully isolated from the living space with partitions. Figure 1 provides two vents to the outside, one within a foot of the ceiling and one within a foot of the floor. If vertical ducts are used to bring combustion air to the appliance each vent should be sized at one square inch of free vent area per 4000 BTUH of the appliance input rating.

(This is based on a half-remembered long-ago conversation with a service person working on my furnace and heater, which are in the center of my mostly-unfinished basement: I mentioned framing in a utility room around them and he said not to for that reason).

The best solution for your climate is to have the poly sheet between drywall and joists but since you already have the drywall up what you propose seems like a good option. For your region the Building Science reports suggest the vapor barrier inside of the insulation.

I have actually seen this done two ways. Up and over every joists, back flat on drywall and back over next joist and taped everywhere. And then I have also seen the vapor barrier laid flat on the joists and taped.

There are pros and cons to each method. First going up and over every joist is a pain to install. It literally takes forever. You might spend a whole day doing this right.

By going flat on top of the joists it is easier. You also benefit from a "neutral zone" that brings basically free r-value. The flip side is if you do have a leak it will probably find itself into a tape seem (opposed to just sitting on the poly on the drywall). Also and the reason I don't install like this - makes it very hard to move electrical or run new electrical in the attic (why I also would never spray foam an attic).

So I am on board with your plan but probably minus the sheeting. If the sheeting is already there then I would attach the poly sheets right on top of the sheeting and make sure it is taped well. I mean you can put up as many vapor barriers as you want on top of each other. As long as the top one is installed right all should be fine.