I think I should leave others to answer that one, cos nearly all bunny herbs die back in winter naturally, & my gardening has always been seasonal, cos I've a cold house, & enough day light is the main factor. They'll usually germinate in temps of about 18C, but tend to go straggly fom lack of light, then die off. Putting a large piece of aluminium foil mounted on cardboard to stiffen it, behind the plant helps to increase light by reflection. Getting watering right is difficult too.
Lambs lettuce naturally grows through winter (no relation at all to what we think of as lettuce) & thyme over winters well. Cheapest way to get more thyme is to buy a small bunch of stems from the supermarket, put them in a cup with 1/2" water & they'll root themselves. when roots appear transfer into soil.
For carrrot tops get a few carrots which haven't been scrubbed, likewise put in a mug of water, & they'll grow roots & start to sprout. Regular change of water helps to stop rot/algal growth.
When the days start getting longer, the cheapest way to get more woody perennial herbs (come every year) eg miniature roses, rosemary, is to take cuttings from 1 plant, get some oasis, cut into 3/4" cubes, put water in to 1/2 way up the cube & plant the whole thing in good soil when the roots come through. the best place to cut the stem is beteen 2 leaves & strip the leaves form the lower part of the stem then dip in rooting hormone (from all gardening shops.)
It doesn't work after midsummer.
I
don't want to put you off. I
do want to save you disappointment. You'll have much better success growing them when they're supposed to grow! They're a doddle to grow then. So I'd say wait a couple of months. It'll make all the difference.
I'd also say sow the seeds thinly - well separated so they can grow good roots. I start seeds off in much wider, shallower plastic trays about 2'' deep, & then move to yoghut pots as they grow. also the plastic has to be pristine clean, or you'll get nasties like moulds, & always put those drainage holes in, even a bit of gravel at the bottom of yoghut pots.