The boots will have one buckle but I wouldn't be to concerned about the brand, I'd be more concerned getting the fit right. 3 yr olds aren't great on feedback. A good shop should be able to help you with this. The same for the helmet. However, if you're buying a helmet, I'd get it early so she has a pretty color and you have a choice of fit. YOU should carry the helmet when it's not on her head since she can damage it if (when) she drops it on something hard. I'm sure you know this, but NOTHING goes under the helmet except maybe a turtle fur shellaclava. The helmet should fit just like yours. It should fit snuggly under the throat and it shouldn't rock back and forth. It will have some give, but not so much that it falls forward or back. The brand will depend on your daughter's head (round, oval...)
Kids have more sensative skin so make sure you have a neck gator for her. She should have mittens and you might want to tuck extra mittens and socks in your pockets since sweaty hands get cold. Dermatone is great for the skin or I used Balmex also (pretty smell). When you go inside, check for moisture and swap out the gloves, neck gator or socks if needed. About the socks...dont' change them until you're ready to go outside again. 3 hr olds have a tendency to walk around the lodge in their socks.
In your ski bag, have extra undies, long underwear and socks. It's cold outside, it's a long way to the bathroom (compared to your house) and it takes forever to get all of those clothes off when she needs to go. Trust me on this, and do this until she's 7. The backup pair probably won't be used and don't need to be of the same quality, but have something handy. I've skied with a lot of little girls and the cold and the clothes spell trouble.
Learning aides...You can get edgie wedgies to keep the skis in a wedge. There is also a harness and there's a pole that I've seen used. If the ski shop doesn't have these, try Reliable Racing. I use to have my daughter ski in front of me with a ski pole holding her in place. Make sure you're holding onto a part of her because kids will slip down under the poles. This of course is murder on your back, today, I'd probably go for the harness and an edgie wedgie.
When you get on the lift, she's either between 2 parents, or she's next to the liftie. Ask to have the lift slowed if you need it. It's better then stopping it entirely. Put your right arm around her and lift her into the seat. Hold her back while you pull down the bar. Hold her while she's on the lift. If you're with another adult, then both parents can take an arm and ski off the lift with her. Her skis will be on the ground, but you'll be holding her up. I usually ski with one pole and I put it infront of the kid when I'm getting on the lift and I hold it there until I get off. The handle part faces the child and I use it as a seatbelt to keep them from sliding out.
When she falls, it's usually easier to pick the kids up about 3 feet from the ground to untangle the skis.
Be prepared to eat/drink/pee often. Be amazed. Be frustrated. Be exhausted And have fun.