As long as the fleece is good and thick, they should be fine, and most WAHMs use good fleece. I once made some for DD out of a thin cheap blanket and they were much less reliable than ones made out of thicker fleece. Most will have an extra layer of fleece in the "wet zone" (the bum and crotch where it can get dampest.) They'll need changing if they feel quite damp, if only a little bit damp they could be aired out until dry and worn again but unlike wool they will get smelly after a bit.
They can go in any wash, they'll be fine at 60 degrees but lower temps are fine too. Washing with nappies and towels will make the fleece bobble up more quickly but this isn't really a problem. It's a good idea to avoid fabric conditioner if you can. They dry very quickly, unlike wool which takes days!
In this hot weather (well, this is our third day of warm summer weather
) I always think fleece might be a bit sweaty for LO's to wear, so a fleece soaker or shorts might be more comfortable for warm days. If you can sew yourself, they are very easy and cheap to make, otherwise there are plenty of WAHMs who can make the to the size you need, or buy a preloved pair. Maybe better to size up if you're unsure, waistbands and leg cuffs can usually be rolled up to fit better.