Maet, you will want to wait until the quilt is finished to wash it. I may be reading it wrong but it sounds like you're talking about fusing the squares then washing them before sewing.
I don't know what type of interfacing you're using, but the knit kind is very soft even after fusing.

I don't know about cleaning the cutting mat - I can't recall ever needing to clean mine except for wiping off the fuzzies stuck in the little cuts.

Donna

Thanks Donna, I think I will wait and have a look for the knit interfacing. I was talking about washing the 4" squares that I have already ironed interfacing on, about 20 squares.

one way to clean those "fuzzies" out of your mat is to use a scrubbie....like nylon net, or those bath pouf things. Also can use a art gum eraser and it will get them out.
Hope this helps.