It's OK to lay in the sun while pregnant.
Laying in the sun or sunbathing while pregnant is perfectly fine and safe and actually the sun will benefit you and the baby.
The sun gives you needed vitamins to stay health so soaking up some sun can help you and your unborn baby stay healthy.
Laying on your stomach while pregnant is safe.
Your baby is protected in the womb and cannot be harmed when you lay on your stomach.
You won't squish or kill the baby when you're pregnant no matter how far into the pregnancy you are as long as the baby is in the womb.
It's fine to sleep however you're most comfortable during pregnancy.
However while it's safe to lay or sleep on your stomach or belly while pregnant it's unlikely you'll be able to sleep comfortably on your stomach as your belly grows, but you can sleep face down for as long as it suits you.
I usually sleep on my stomach but when I got farther into my pregnancy I could not sleep comfortably on my stomach.
But if you're comfortable sleeping on your stomach you can do so.