Does it really matter when to replace a carpet in basement, summer or fall/winter? The basement isn't really damp, but probably there are some other arguments/reasons to consider a certain season for such work?
Around Xmas and January. Generally new carpet lines go out in the spring. So in December and January we have found that we can get carpet much cheaper and the installers aren't busy at all.
Also don't let them glue down your pad or carpet. You need to be able to move your carpet if you have a water problem. (good installers would never think about glueing down in a basement but I see it all the time)
Nope. If moisture in your home isn't managed throughout the year you'll have problems. It doesn't matter when a project like that is completed.