In the cases mentioned above in your question I would replace but with a semicolon.
I don't like this picture; do you like it?
I don't know the way to the place; do you know that?
I don't want to cooperate with the person; what about you?
And if you don't like semicolons or think that they are weird, you can always replace them with periods.
I don't like this picture. Do you like it?
I don't know the way to the place. Do you know that?
I don't want to cooperate with the person. What about you?
I think the usage of 'but' in compound sentences in your case is not very good, because we use 'but' when the parts of the phrase are in some kind of strong opposition. E. g. 'Mary doesn't like the ice-cream, but I do.'