Alright, here we go.....instead of old wive's tales and people's personal feelings, let's go with real medical advice....
The appropriate amount for most infants to drink, in ounces, is their weight, multiplied by 2. If the baby weighs 8 pounds, she should consume 16 ounces of formula in 24 hours. 10 pounds is 20 ounces, 5 pounds is 10 ounces. This is the amount of formula that your infant, on average, needs to survive and thrive. If your baby, at one week, is the average 8 pounds, and is eating every 3 hours, which is 8 times per day, she needs 2 ounces per feeding. You'll have to insert your own values here, since we don't know how much your baby weighs. If your baby still seems hungry after her feeding, do offer her more formula, but do not try to encourage her to eat after she seems full. Babies this age do NOT overeat! They simply consume as much as they need.
Most products created to cleanse the mouths of babies are intended for children over 3 months. It is fine to begin using these products at 3 months, even if your baby has no teeth. This will get her used to having you clean her mouth, and when her teeth do come in, it will make life easier. However, when you begin feeding your baby solid foods (i.e. anything other than formula) or juice, you should definately begin cleansing her gums and/or teeth at least once per day, and twice would be better.
Your child should sleep in a crib, bassinet, moses basket, cosleeper, or play pen from her first day home. There are many reasons for this. The first is that children who sleep alone, on their backs, are less likely to die from SIDS. The second is because your child needs to learn to self-soothe, and fall back to sleep on her own if she awakens and is not hungry. The third is because if you let her sleep with you now, it will be exceptionally difficult to break her of it at a later date. Put her in her own bed now!! If she seems to cry because of the large space, swaddle her, or purchase one of those little wedges that goes on either side of her to "hold" her in place.
There are many ways to burp a baby. One is as you say, with her sitting up, but this is difficult for most new moms and young infants, as they cannot sit up well. Try placing her over your shoulder and patting her back. Also, you can lay her, belly down, across your knees, and pat or rub her back. If your baby does develop gas, you can lie her on her back and gently press her legs up, knees against her tummy, and "bicycle" her legs, to help relieve the gas. Also, symethicone drops are sold in every store with baby products, and are not harmful, used up to 12 times per day. You might attempt using a concentrate or ready to feed formula, instead of powder. These are sometimes less likely to cause gas.
Good Luck with your little one, and do feel free to post any questions you have here, everyone is eager to help!