It depends on the day. Sometimes you may feel like buying her ice cream.......then do it! If you don't feel like it then I would say something like, "No, I don't want to spend my money that way today, but if you would like to spend your own money I'll be glad to take you over there."
When she complains then gently (and as sweetly as you can) remind her that when she is all grown up and working she will be able to choose what she spends her money on, or save for what she wants now.
On her birthday I would give her a choice. Would you rather have presents or is there somewhere special you would like to go? That sort of thing. This way you keep in budget and she gets to choose what is most important to her.
We did allowances as you are with our kids. It worked great! On vacations we always gave them extra money that they could choose to spend however they liked. If the whole family was going to play miniature golf then we paid, but if they wanted to play games, etc.... that was on their dime.
All kids complain they are bored. I'd suggest summer camp, trips to the swimming pool as often as you can and invite kids over. She is at the age where she now wants to be surrounded by friends every day.
Try not to be angry when you tell your daughter "no." Try to be sympathetic. "I'm sorry honey, but we don't have plans to go anywhere today. Why don't you call a friend to come over."
Oh.....one more thing. When she asks for something you would like her to have but you think is too expensive you can always tell her how much you are willing to spend. Then she can decide whether she wants to throw in the rest.