If I drop a pair of mountain shoes with the cleats on in the washing machine are they likely to get damaged? Is the machine likely to get damaged? Should I throw some rags in - or even wrap the shoes in a rag? Or is hand washing always preferred?
I suggest a completely cold wash if possible, or as cool as the machine will go, and a gentle cycle. A woollens cycle, for example will throw them around a lot less than a cottons one, preferring soaking. Both shoes and machine should be fine but it can get rather noisy.
Sports shoes (including cycling shoes) are usually synthetic and have no problem being wet - after all, we ride in the rain.
There are products that will kill bacteria even in cold water. The one I use certainly seems effective on jerseys, base layers, etc. in completely unheated water - probably about 10°C in winter. There are also laundry disinfectants that get used in the rinse, if you choose a detergent that's better on the grime. Note that a lot of oily dirt sticks very well to synthetic fabrics, as you'll know if your jersey has ever got chain grease on it. You might not get that sort of dirt off.
If you have cleats that stick out and can't remove them, I suggest it might be a good idea to cover those. You could, for example, make an old towel into a bag, either by sewing or by using safety pins. An oversize pair of thick socks might even stretch enough.
Do try to dry them quickly, but without huge amounts of heat. At least hang them in a breeze or in front of a fan. If they couldn't handle being left in the sun they wouldn't be fit for purpose, so that's a good option, but needs airflow as well. Personally I have a couple of old computer fans duct-taped to tubes for drying shoes and gloves with room-temperature air.
Maybe. Contrary to popular belief, leather is not immediately destroyed in washing machine. It just needs long time to dry afterwards and some conditioner because machine wash removes all oils too. The risks are that high water temperature may damage plastics and glues, and low temperature does not remove dirt and especially bacteria.
I would remove cleats, laces and sock liner just to allow water circulation. For what it's worth, I have tried machine washing leather shoes from Shimano that I was about to throw away. Both the shoes and the washing machine survived fine, but I wouldn't try it on shoes I would not risk.