I would make a list of the herbs I like and take it along when shopping for seed. While at the store, I would read the instructions on the seed package.
The link contains a list of several herbs and when to start them. There are also links to "How to".
HTH

Here is a link that might be useful: List

How cold does that room get? Few seeds will germinate under cold conditions, and depending upon how cold...some of the developing plants might not be too happy, either.

If you could provide heat to the germination medium (in the way of cables or a heat mat), then the transplants could be moved to the sunroom for growing on, as long as the temperatures aren't too awfully cold.

I have some herbs just poking up good and they are in 3" terra cotta pots. I got the "kit" from Staples in the clearance section at back of the store, and the brand is the same as the Chia Pets.

They said to put the pots on top of the fridge, which is always a source of warmth. Now I have them sitting on top of the gas water heater, and turn them about twice a day.

One of the most rewarding herbs to grow is the basil which grows quickly. Then I'm happy with the looks of the dill and the parsleyand the thyme. But I'm disappointed in the chives. I have better luck buying a pot already started. Rosemary is pretty to grow, but does not like wet feet. Very touchy to get it right with rosemary. My rosemary is outdoors in the flower bed with no protection, and it is doing great.

For Christmas, my DIL gave us a hydroponics set up with auto feeding of 3 kinds of herbs. Again, the basil is exceptional, quite big enough to start pinching off leaves.

When we redo our kitchen, the window over the sink will be replaced with a garden window for my herbs and smaller potted plants. It faces south, and by the time the hot weather arrives, the roof overhang should give this area a good bit of shade. Right now, it is a perfect spot for house plants I brought indoors for the winter.

Here are a couple of pictures:

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If the fridge top works for herbs, it also works for birdnest fern and staghorn fern and nun orchids.
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This is the hydroponics thingy. A lot of trouble IMHO.
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Thanks for the pictures. You have some healthy looking plants! I'm going to try basil and dill, two of my favorites.