Yes, you can wash sneakers in the washer. Delicate, cold, small amount of detergent, laces out (Dan!) and in a laundry bag or the pair entirely in a bag laundry bag.
You can wash with other sneakers but I wouldn’t wash with clothes.
Consider getting a boot/mitt/hat dryer (like the ones at Costco). Very handy for quick drying and no mold, bacteria, fungi stink. I use mine all year round for this and weather related needs.
Note that the fabric on the soles might let go over time. The glue isn’t so strong.
Reminds me of an oldie:
“Roses are red, Violets are blue. Some poems rhyme, This one don’t.”