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.