Getting to stay home alone
Your child's individual level of maturity and responsibility play a large part in determining when they can be left alone at home, and for what amount of time. I've met 10-year-olds who are more responsible than their teen-age sibs, and are therefore safer bets to remain at home without parental supervision. In addition, your community will most likely have ordinances or policies about the minimum legal unattended age so you'll be wise to check on that.
In general, though, I believe that it can be safe for an 8- or 9-year-old to be allowed home alone while you quickly run to the convenience store (10 to 15 minutes or so) or to do a quick errand. That's assuming that you have your cell phone with you for emergency contact, that the child can be trusted to stay inside the home without answering the door or letting friends in, and that the telephone has caller ID so that it's answered only if he or she determines that it's a family member calling. Otherwise the call should be allowed to go to the answering machine without the child picking up.
Being home alone after school or during the summer for extended periods of time is a horse of a different color though. Consider setting up an arrangement with a neighbor to watch your kids as well as hers while you are at work, or to place your children in a day camp situation if this is possible. Unsupervised children tend to become bored when left alone, break house rules, leave the premises, let others in, or find their way onto the Internet when parents are not on patrol.
By the early teen years, though, many kids are responsible and mature enough to follow house rules and to be allowed to stay home alone after school or during the summer. But, know your individual child — impulsive kids often act before thinking and wind up in trouble. If you're going to be worried while at work, it's just not worth it. Even though Junior might love the freedom and flexibility of having the house to himself, if you can't trust his judgment, don't do it. It's better to put up with some whining and complaining when you schedule him for yet another summer of day camps than to have to worry why he's not answering the phone and you can't leave work to check on him.
Getting to wear makeup or shaving legs
It's not unusual for girls in the fourth and fifth grades (10- and 11-year-olds) to show a distinct interest in wearing makeup and shaving their legs. Although most preschoolers love to play with Mom's makeup as a form of "dress up" or to pretend shave with an empty razor, many begin to seriously push for these privileges in the later grade school years. Most parents, though, believe that middle school is the proper time for these grown-up behaviors to begin to be allowed, and many dramas have resulted when the kid's desires conflict with the parent's expectations. My suggestion to clients is to try to reach a compromise, based upon the average age that other girls in your child's environment (school, church group, and neighborhood) are allowed to engage in these behaviors, your personal values and age when you first shaved or wore makeup to school, and the privileges allowed to your child's intimate group of friends. It's a tricky balance trying to allow your child to "fit in" with what her friends are allowed to do while at the same time staying within the boundaries of propriety. Personally I feel that shaving legs in fifth grade is reasonable, but be sure to warn your young lady that once she begins shaving it tends to be "forever". And, it hurts! Band-Aids on the knee are less than attractive and she might wish to put this tedious chore off for several months or a year in order to avoid that inevitability. If you do allow the razor, you might wish to purchase an electric model initially so that she doesn't look like a pin cushion. Show her how to do a good job and consider this a bonding opportunity if nothing else!
Wearing makeup is somewhat trickier, though. Tweens and young teens have the tendency to want to wear gobs of blue eye shadow and bright lipstick. My clients have had the best luck when they've taken their daughters (often with a friend and her mother) for an application at the makeup counter in a department store. Make an appointment for the girls and suggest to the sales person that the more "natural" the look, the better. Toned-down shades are most appropriate and often a barely colored lip gloss does the trick! Have the expert discuss the concept of "less is more" when it comes to makeup for young ladies, as well as the importance of keeping their skin clean and not sharing products with others for hygienic reasons. Colored lip gloss and light shades of lipstick are often acceptable in grade school (if this does not conflict with school policy), but eye shadow and mascara should be saved for the later middle or high school years. And, your child should have to pay for these products herself--take advantage of this as a teachable moment — that adult-like privileges (makeup) come with adult-like responsibilities (hitting her own piggy-bank).