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.