Modern codes require a fire rated door between garages and living quarters. That would be the first thing to factor in if replacing the door.

Sounds as if building a custom frame and hanging a proper door is the best option, especially since you don't want a threshhold. I take it there is not a water intrusion problem in the garage and subsequentially into the house?

Preventing water intrusion is the major reason for threshholds.

You will still need a sweep on the bottom of the door to insulate. Sweeps almost need to be built into doors for longevity and decent operation, rather than added on one side after installation. If you can buy a correct slab door with a built in sweep, that eliminates having to do the necessary machinihg.

Building a custom frame is not a first time DIY job. It is not an impossible first timer job, but doing it usually becomes much more expensive than hiring it done.

But, hiring it done means finding someone with experience and ability---which don't necessarily go together.

You have made a series of incorrect statements and based your plan on these statements. A prehung door will not necessarily fix the problem. It all depends on what width jamb you need. They don't make them all.
Fire rated doors do not have a wooden threshhold. It would not make sense as the wood is combustible.
Go to the local lumberyard with your dimensions and the room setup and let them guide you.
Then come back and confirm what they said.
Or... go to another lumberyard and see if you get the same info.
Ron

If you buy a pre-hung unit, another option you might consider is a low profile "handicap" sill. Again check with your local lumberyard or building material supply, (find out who there is most knowledgeable about >doors
, they will have several suggestions.
Should you decide to hang the door yourself, here are a couple of thoughts:
Most of the heat loss will be from the glass and air leaks. Be sure and weatherstrip all around the frame to prevent air infiltration. You can get new insulated glass from a local glass company and replace the existing glass. An insulated curtain or blinds on the inside can also help.

A door to a garage is not allowed to connect to a sleeping room and it would only need to be fire-rated if some unusual code applied (like the UBC or NFPA 101, etc.).

The typical residential code requirement (IRC and its predecessor CABO, etc.) is for a solid wood, a honey-comb or solid core steel, or a 20-minute fire-rated door so it is usually possible to use an ordinary solid core wood door with an ordinary wood frame without rated hardware and without an automatic closer. (If you use a 20-minute rated door the frame and hardware should be rated/labeled/listed as well). It is always wise to ask the local building official; I've run into some who also like to enforce NFPA 101 but some aren't concerned about the pairing of a rated door and an unrated frame.

These doors are not usually located in fire-rated walls (usually only the finish on the garage side is required to be rated board, if rated at all) so, making the doors carry a fire rating label from a testing agency makes little sense. Typical garage separation walls are, in reality, smoke partitions and the doors should be considered smoke doors. Therefore a more pertinent requirement would be for door gasketing which some jurisdiction do require.

The risk of fire in a garage is far less than in a kitchen or bedroom. The requirement for a quasi smoke partition and door are due to the fact that a fire might not be detected as quickly in that location as in occupied parts of the house.