Are my pepper plants doing okay?

I have a container of young bell pepper sprouts, which I assumed were doing fine until I saw a time lapse video of the plant growing (posted below). The description for this video states the footage was taken over 8 days, which makes me wonder, is something wrong with mine? Mine are currently about an inch tall and this has taken them about 2 weeks since first appearing at the soil surface. Is this normal to be so slow? I keep them watered and always in heat - during the day, full sun and at night on a heated surface indoors. They look good and healthy but apparently their growth is slow compared to others.

The plants in the video are stretching because of insufficient light. You mentioned yours are getting full sun, which is fantastic and what they need. They appear to be growing slow because after a seed germinates, most tend to "pause" in appearance of above ground growth, and what's happening is all the plants energy is being devoted to root growth. Just wait, and soon you'll see nice compact leaf growth.

Wow, I did not know that. Good to know and a relief too. Took me a long time to successfully germinate the seeds, so I was really hoping I hadn't lost them.

Another question I forgot to ask. Should the soil dampness remain the same as when germinating the seeds or should it become less once sprouted? If the same, would root rot be a concern?

Go ahead and skip watering a day or two (or three). overly wet conditions slow things down, and can pose a threat of things such as root rot. Let it "dry" out a little but still remain moist. I've noticed that peppers in particular, which I also grow from seed, don't like wet feet.

Sometimes seedlings get a little spindly when they are either not getting enough light or the light they receive is too far away ie. indoor grow lights are positioned to high from the seedlings forcing them to stretch and become elongated. I would be sure your giving them at least 8-10 hours of full sun or exposure to an artificial light source. Also, don't water too much which as this could lead to dampening off:

https://www.planetnatural.com/pest-problem-solver/plant-disease/damping-off/