"Could you wash the dog?"

Is not a question asking whether the person is capable of washing the dog. It's a command phrased politely.

"Try to wash the dog"

"Try and wash the dog"

If you had no prior information on whether the dog likes water or not, I'd say that the try-and version expresses a greater level of confidence that washing the dog will be successful, in other words it's a command.

Whereas try-to could be read either straight (this task may fail) or as a command phrased politely.

It still implies possibility of failure, but in the example of the commenter above, that possibility is almost low enough to the point of expectation (but not quite) and "try to" would increase that possibility in the direction of failure. Nuance!