If the verb after wish is an action verb, use 'would' or 'could' instead of the past simple. It refers to the future.
(Everyone changed out of their uniforms to casual dress except you)
I wish I (had) changed clothes.
-This refers to the past, and your regret that you didn't do something.
(It's hot and you're wearing sweats)
I wish I could change clothes.
-This talks about the future and refers to the action of changing.
The previous examples you gave all described states.
I don't have a dog - I wish I had a dog
Today is Friday - I wish it were Saturday
I speak English - I wish I spoke French
I can't drink milk - I wish I could drink milk
So any verb that is used as an action requires 'would' or 'could.'