Yes you may mow your lawn at this time of year. Long grass can suffer winter kill more than a short turf.
Avoid standing on frozen grass as this will cause cell damage to the plants resulting ugly brown patches or footprints in your lawn.
It is favourable to mow when the ground is dry otherwise things can get a little muddy.

Ryegrass seed can be oversown from late February onwards
Finer grasses such as bents and fescues should be left a little later as the seed has less vigour a requires a little more temperature to germinate.

You could apply some grass seed now but there is the danger that should there be a prolonged period of cool, wet weather the seed or small seedlings could rot in the ground before they become established.

If you are in the southern counties of the uk and on reasonably free draining soil I think you'd stand a reasonably good chance of a successful take using a dwarf perennial ryegrass, although it wont be very quick to establish.

Grassman