Suaz I think you are worrying for no reason, in fact your plants sound (and look) healthy to me. White fungal-looking fibres is part of normal root growth.
Damping off is a dramtic falling over of seedlings where the stem starts to rot quite quickly and seedlings can turn to almost invisible mush within a couple of days. It is caused by excess humidity (really excess!) and actually happened to some of my spinach and beetroot recently, where I had fleece on them in recent cold weather, on my new hotbed in the greenhouse. I had not appreciated the amount of humidity rising from the heap! Luckily everything else is fine after removing the fleece.
Trays which have had damped off seedlings in them are not ‘infected’. As for potato blight, it is the prevailing conditions which count, not (supposedly) existing spores. Even if they survived the trays drying out, which is unlikely to say the least, they still need perfect conditions to infect plants.