It is not designed for inverted placement. But it could be achieved by running part of the power cord outside the casing which would entail cutting some entry grooves in the case. The major problem would be stopping the weight of the Echo Dot causing possible disconnection of the power plug.
Yes sure can, bottom, top and each side, but I wouldn’t recommend side because the weight of the echo would put a strain on the mounting sleeve.
You may have to turn wall switch on before plugging it in
Yes, I have done exactly the same so that the echo dot leans backwards and reamains on top since I have plugged it into a power board which is placed on the top of a shelve. You just need to slot the adapter into the socket accordingly.