Hi Sasha. For the second sentence, both are fine and gramatically correct although in my opinion, the way you phrased it the second time sounds a little better. As for the first sentence, again, both are gramtically correct but the implied meaning does differ a bit between them. 'Have you ALREADY been to the supermarket?' - this sounds like I suspect that the answer is likely to be yes because i know you had planned to go. Maybe i wanted you to buy something for me and I am checking with you whether or no it's too late. "Have you been to the supermarket yet?" - if I'm asking this question, maybe i knew you had planned to go but I really have no idea whether you went or not.

Regarding 2. either are OK and natural. Regarding 1, "yet" and "already" have slightly different meanings. For example, you said you would go to the supermarket at 4pm but at 2pm I see you with some shopping bags so I ask : "Have you already been to the supermarket? I was expecting you to go later." Alternatively, it is 430 and you are still at home. I am not sure if you have been to the supermarket or not. So I ask "Have you been to the supermarket yet?"