Fleas walk once they're on their host. They jump to get on to their host, or to get away from your fingers, but otherwise they walk (presumably because jumping when you're on a host makes it quite likely that you'll land *off* the host)

To my eye they do look pointed at the rear, but it depends on what you mean by pointed.

Dark brown and shiny is right for flea.

Are you finding any blood rich faeces? ie do any of the little black bits in bed turn into red smudges if you dampen the sheet around them?

Fleas tend to leave lines of bites, usually on warm bits of your body. I wonder whether they actually bite at night, or whether they bite all through the day, but it's the warmth in bed that makes the bites itch.

Bed bugs, according to google, will bite on faces.

Whatever they are, think it is highly unlikely that they will move away when the weather turns. You need to get them identified and sorted as soon as possible. Fleas are easier to get rid of than bedbugs.