Можно ли поставить в present continuous глагол to stay?

Да, можно. даже группа есть "staying alive", что в переводе значит "остаться в живых".

Дэ am/is/are staying

11. they went to st. petersburg in summer.         where did they go in summer?         when did they go to  st. petersburg ?         who  went to st. petersburg in summer ?   12. i'll listen to rock music at home.         who will  listen to rock music at home ?         where will you listen to rock music ?         what kind of music will you listen to?   13. mary is going to learn english.         who  is going to learn english?         what language is she going to learn ?   14. sam and his girlfriend have come to stay with us this weekend.        who  has come to stay with us this weekend ?         where have  sam and his girlfriend come to stay ?   15. there were five students in the class.       where  were there five students ?       how many students were there in the class?   16. it usually takes me five minutes to get to the college.       how long does it take you to get to the college?   17. my father is forty-seven years old.       who  is forty-seven years old ?       how old is your father?   18. this magazine costs fifteen rubles.         how much does this magazine cost ?         what  costs fifteen rubles ?         what kind of magazine costs  fifteen rubles ?   19. they collected stamps.       what did they collect?       who  collected stamps ?   20. we have been friends with vlad for quite a long time.       how long  have you been friends with vlad ?   21. it is raining.       is it raining? 22. she likes cooking.       does she like cooking? 23. he has taken a few pictures.       has he taken a few pics? 24. they don't know english.       do they know english? 25. we didn't go to a disco yesterday.         did you go to the disco yesterday?   26. she was my great friend.       was she your great friend? 27. mike collected posters last year.         did mike collect posters last year?