Yes its perfectly legal. However, I say that you should hang it on a pole instead.
Check the HOA rules if you live in that kind of community. Then hang the sign in the window most easily seen by somebody walking by with a dog. Post your phone number.
Just do it. Sometimes it is easier to ask for forgiveness than to ask for permission, if your area has a HOA may have rules against it, once again just do it if no one complains then it will be ok. Some stores like grocery stores have a bulletin board where you can advertise services like yours, you can also advertise on craigslist.
Sure, if you parents are ok with it.
No
I'm doing dog walking for the locals and is it okay to put a sign in my bedroom window to get attention for it? im only doing it to get some pocket money, im 15