Yes, a pretty big difference. Mouth like south but mouse like house. My house is south and not north. My mouth can be both opened and closed.

Yes, they are different. I suggest you to learn more about the "TH" pronunciation and so you will notice the difference. The difference is the way you place your tongue into the mouth.

No. They are different. The end of MOUTH sounded like T. some dictionaries have voice. You may listen to them for distinguishing.

check another clearer souce where you listen to the pronunciation of these or similar words. In my view the unvoiced Th sound is more of the F sound with a *little* tinge of S like.

I see you are a French speaker; there is no "th" sound in French. You must learn to put the tongue out a little to say this sound correctly. Welcome to English.