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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.
Oh god, me too. Fixing that and how having it connected to my TV disrupts every other device connected to it for some reason (HDMI-CEC problems maybe).