IMO they do need warmth to germinate, mine start on the windowledge, near a radiator, they never germinate or grow well, when they're cold and damp.
They do always look unhappy/weedy and lay down, while they are small, I often snip the black seeds casings off 'cos they seem to weigh them down..
Once they're pricked out into modules, and grown on a bit, they usually do straighten themselves up.
Were your seeds new this year, unlike other veggies the seeds don't seem to germinate unless they are fresh. I had a packet last year which was a year old, still within date according to the packet, and only a few germinated - so I feel your pain