I have no experience in growing in a cold climate. Best to wait for some responses from lower zones. But in case you don't get any, then here are my thoughts:

Basil needs heat to grow. I won't say give up, but I reckon at most you're only going to get it to survive. Unless you can give it heat as well as light, then it's not going to grow. I grow basil outside in winter where I live, in average temps of about 10°C. But it doesn't really grow; it just sits there dormant. I keep some potted basil plants in a small poly tunnel. They do grow in that, because it heats up during the day and retains most of it at night.

Seedlings like that, where the stems are still tender and they only have a few leaves... If you do it again next year, I reckon start growing earlier, so that the plants are stronger and the stems have got a woody exterior before going into winter.

I suggest giving it as much sun as possible during the day, but take it away from the window at night and put it near the heater or something.

You've actually got several plants there. I'd suggesting splitting them up to
maximise your potential success rate. If you can get some more of those
containers, that size is fine for each of the stems. But keep in mind
that the smaller the container is, the more exposed the roots will be to the
cold. You don't want too big of a container though, because that will
make moisture levels harder to control. A small planter box might be
better, and space the stems a few inches apart. Or you could find some
way to insulate the pots.

Get a good-quality potting mix. Keep the soil lightly moist. Not wet, not dry. If they don't grow then they won't use up much water. Just check the soil beforehand. I don't fertilise in winter. Any growth that comes from fertilising will be really vulnerable to the cold. I suppose it depends on how warm it is inside your house.

If you can get it to actually grow steadily, then clipping a stem every now and then won't hurt. But if not, then every leaf that it loses is a loss of potential energy that it needs to survive the winter.

Good luck. Whatever the result, I'd be really interested to hear how it goes. There was a place in Yellowknife that I stayed several years ago (in
summer) and they had a big herb patch and greenhouse. I have no idea
whether they tried growing basil in winter, but I'm super interested
to know if it's possible.

Growing basil indoors in the winter up north is easy. In order to have success, though, there are a few things to consider. 1. Lighting. Basil prefers full sun and requires more sun than your window will provide in the winter. I suggest that you provide artificial light for it if you want it to thrive and be productive. A cheap 2light shop light with one warm and one cool light tube will provide the spectrum that the basil needs. Also, leave the light on for 12-16 hours a day to provide the plant with enough light and place the light as close to plant as possible without it touching the plant. Reflect light back on to the plant if possible (I use aluminum trays from the dollar store). 2. Transplant the plant into the next pot size up using a soil-less potting mix. When the plant sends roots out the drain holes repot into the next size pot and so on. Do not use potting soil-use potting mix. 3. Bottom water when needed-when the top of the soil dries out and fertilize with an all purpose fertilizer once a week at the recommended rate (on the package). Miracle Grow or similar will work perfectly. Basil is a hungry herb and needs fertilizer to develop those delicious leaves. 4. When harvesting make sure to snip off directly above a set of leaves with a clean tool(I use scissors). 5. Enjoy! I hope this helps you!