OK- there are a few issues here that I can see. You are right- this is your daughter and you have every right to decide when to intro foods and such. Your mom, having raised a child or two herself, is probably trying to be helpful and has your and your dd's best interests at heart. I would sit her down, talk to her, and tell her that you appreciate her concern, but that as long as you aren't putting your child at risk, you would appreciate more her support in your making your own motherly choices.
As for sippy cups- I have a strong opinion on those, and as your daughter's mom, please feel free to disagree. I work with infants and toddlers in a home visiting program (Early Head Start), and I recommend you skip sippies altogether for a couple of reasons.
1. Language development. In order for children to learn to speak properly, they need to develop certain muscles in the mouth. This happens when they learn to drink from a cup- not a sippy. A sippy, if over-used, will actually cause speech delays because it is so much like a bottle. Your baby can learn to drink from a cup now- I started mine at 4 mo- as soon as I started solids. It's a messy process, and you have to hold the cup for her until maybe a year, but it can be done.
2. Teeth. Tooth decay is caused by a bacteria. Bacteria feeds on sugars. Juice, milk, formula all have some type of sugars. With a sippy, a child can take a few sips, leave it, take a few more sips 10 mins later...this gives tooth-eating bacteria a constant food supply, which can cause lots of problems now, and can also affect permanent teeth.
If you still choose to go with a sippy, put only water in it, and don't use it as a replacement for the bottle. Skip juice altogether for now- baby doesn't need it at all. The American Academy of Pediatrics (AAP) recommends no juice before age 2, simply because there's not a whole lot of benefit from it. Give her real fruits instead.
And the bottle and foods- dd should be still drinking around 32-40 oz of formula daily, give or take a few oz depending on how hungry little one is. It's ok to give her fruits and veg with cereal- I assume 2x a day right now? Cereal and fruit for breakfast and cereal and veg for dinner? That's really all she needs along with formula. As long as you are introducing foods slowly (1 each week or so)- that way you will know what caused it if she has an allergic reaction. Also, please be careful to avoid because of allergy risks- nut products, milk, milk products (cheese and yogurt can come at 9 mo. or so), honey (can cause botulism poisoning), berries, citrus fruits (including tomato and tomato sauce). Wait until she's at least one year to start these foods. Nuts and peanut butters should wait until 3 yo because of choking risks (as well as raisins, hot dogs, popcorn, hard candies, gum, etc.).
As for second group of Gerber foods- just be sure she can take the texture of it ok. You'll know because she'll mash it around in her mouth before trying to swallow it.
I hope I didn't come on too strong- it's ok for you to ask people when to intro new foods to your baby to be safe. There's so much info out there to try to keep track of. Your mom may or may not know what the latest info is, but your doc may be very helpful with the latest recommendations. Or, please feel free to contact me offlist if you have ANY questions. Remember, she is your baby, and you can choose to raise her however you'd like. But it's also your resposibility to make informed choices. Trust your instincts in the end, but get the info in the beginning. Good luck!
PS- It's ok for baby to not hold her bottle at this point. You want to see her put her hands on it, but don't be in a hurry to put her down with it. She could choke, inhale it into her lungs, over eat, and it's best for her to be held and cuddled during this so short infant stage.