Is ‘a dog which is barking’ synonymous with ‘a barking dog’?

I am wondering whether a barking dog means the same as a dog which is barking.

Here is the saying a barking dog does not bite. Does a barking dog herein mean a dog which tends to bark in face of threat or a dog which is barking?

Similarly, could I saw a patient coughing be rephrased as I saw a coughing patient?

Besides, does a waiting car refers to a car that is arranged a priori to pick sb. up or a car that happens to be seen waiting, say for passengers getting in or for a traffic light turning green?

Also, sleeping as an adjective collocates with child, as in this ODE example a sleeping child; could one likewise say a sleeping student/lady/man/etc.?

Thanks in advance!

Both are probably correct, but being pedantic I’d argue they can have slightly different connotations. To focus on one of your examples,

I saw a patient who was coughing

This implies that the patient was coughing when you saw them. You could argue however that this also has an implication of being temporary.

I saw a coughing patient

While this also implies that you saw a patient who was coughing, one could argue that in this context, it becomes a more permanent attribute as if they were known for it. “Oh, don’t worry about him, that’s the coughing patient. That’s normal. “