"Putting" can be used as a verb though generally it is followed by "on". For example "I am putting stickers on my computer". "She is putting stickers on the wall". Technically you can also say you are "sticking" stickers but that can sound strange because you are saying two similar words in a row. You also just say that you are "playing with stickers" as you did above. You could say "placing" or "affixing" stickers but that is less usual. In my mind it implies that you are sticking stickers in specific places (with thought behind it).

Usually,we use "put sth on...."In that sentence,u can say my daughter and I are putting stickers on the wall,this is a complete and correct sentence.