If dampening off is a consistent problem it would be worth investing in some vermiculite. It's not too expensive and you only need cover the top of the compost (try ebay, this was a good deal -
http://www.ebay.co.uk/itm/370919248724?var=640134327020&ssPageName=STRK:MEWNX:IT&_trksid=p3984.m1439.l2649). I've had years of success with covered propagators, but have found that some seedlings do suffer from dampening off more than others - usually the smaller seeds in particular, with tomatoes being a prime candidate. Our house is not the warmest either (more so since moving, it's bigger!) so I only ever start things off in the latter part of the suggest 'range' of months to do so (we're in West Wales).
What compost do you use? That can make a difference too. I tend to use multipurpose organic and peat free, but whilst it's fine for more robust things like squash seeds, again, things like tomatoes don't seem so happy in it... not that it bothers the 'volunteers' that seem to come free with the content, they are. clearly, made of stronger stuff!! Try a compost specific to seed growing - they are often sterilized, which helps, and there are organic versions too.
As for when to uncover - cool is definitely better than too hot, so start with turning the heat off once they get their first full true leaves, but leaving them covered for a week or so, then take the lids off during the day and back on at night and so on. Judge by how they grow, if they've got this far they want to keep going, just make sure they have plenty of light on all sides by turning them regularly if they are on a windowsill (I'm sure you know a lot of this, just want to give a fullsome answer!).
Hope they do okay