You can buy weed supressant membrane that will let water through, I havent heard of visqueen.

Why would you put it under your lawn......I may be quite wrong but I haven't heard of that being done

I presume you're trying to stop perennial weeds coming back through the lawn organically.

Unless you pricked a huge number of holes in the visqueen (which i believe is used on building sites as a water proof barrier)I'd say don't do it it. In fact I certainly wouldn't do it at all.

I'm not even sure a proper woven weed proof membrane is a good idea under a lawn unless it would covered with a substantial depth of new topsoil .Once again I would not do that at all.

Personally I'd spray the weeds with Verdone Extra and then lay the lawn but if you're not happy to spray then an organic approach would be to individually dig up perennial weeds. Annual weeds are not a problem in a lawn as they'll die with repeated mowing

This is not a good idea, I'd follow anchorman's advise and get rid of perennial weeds by spraying or hand weeding, then lay your lawn.