Not an expert, but I start my easy ones, like tomatoes, in the greenhouse without any problems. I have water barrels under the bench, so that kind of supplies bottom heat. The hard to germinate stuff gets started in the house then moved.

I start mine in the greenhouse on heat mats, and get very fast germination. However, if you are starting them now when you have shorter days (and possibly overcast) they may get pretty leggy competing for the reduced sunlight. I keep mine in the greenhouse until the weather is nice enough to plant then out, and this also helps keep the slugs and other pests off them when they are small. There are other approaches...this is just what I do.

Much depends on your operation. I always figured that I would not want to heat a greenhouse when I'm already heating my house and until late Feb. I usually have enough room in my house (although I typically hang 28 shop lights to provide light). I will admit that artificially lighted plants will need to be hardened to the greenhouse. Shade cloth helps with that adjustment.

I really like my heated basement (where coal stoker sets)for germinating plants. IMO heat mats a a waste of money when you have a room that typically stays 80-85F. I also know that in my house I won't have pest issues on the scale of a greenhouse but then all other house plants must go once vegetables germinate.

For my small operation:
With me it is all a matter of time and temperature. If GH is cold, I start in house with or without heat mat. Then to GH as soon as I can for the light. If I have to keep then in the house, they need lights as soon as they come up. Heat mat goes away as soon as they come up. My GH gets pretty cool at night here at high altitude so I like starting inside where I can hurry them up and baby them. We get late frost here so the GH is a great help in starting early.