1. It is highly unlikely that your apartment will not have shutters. Even if you have a sliding door to the balcony, there will be shutter type sliding door as well.
2. Start taking them when you like really, the odourless type of course. But I would say a week or two before you go. They are not normally recommended for children under 12 years, so I would say it should be OK, but do check with your pharmacist when you purchase them. I will ask my parnter when she comes in, she is a Pharmacist Dispenser Supervisor, and get back to you.
3. The Avon product referred to on this topic is, Avon Skin So Soft, and is highly recommended. However, if you child is allergic to skin products, then as described in an earlier post, she could where a patch on her wrist, or attached to her clothes, changeable daily.
4. Lavender, I have not heard of personally as a deterrant against insect bites.
You can also, if your child likes it, get her to eat Marmite on toast or bread for breakfast or on a daily basis, this again is a valid remedy against the bites of mosquitos.
If you go back through this thread, and others also on this site, you will find many alternatives against the dreaded mosquito. I hope this has helped you in someway.