From personal experience, I think that it's okay to let the baby nurse to sleep (I can't speak for bottle-feeding) for the first few months. It's extremely comforting to them, good bonding time for you, and yes, very natural. At some point, though, preferably before 1 year, you should start letting her learn to fall asleep on her own. It IS an important "skill" for a child to learn, but contrary to the beliefs of "sleep trainers" it DOES NOT have to be done in the first few months. A child can learn self-soothing at any stage -- it's just that it's easier when they're smaller.
That said, I did let the nursing to sleep go on for too long (for my personal comfort) and it DOES get harder to teach self-soothing the older your baby gets. I would say sometime after she is eating solids and doesn't need the calories from nursing, it's good to start pulling her off the breast (or taking away the bottle, though I have no real knowledge of bottle-feeding) when she's starting to get sleepy, and put her down to sleep on her own. Just understand one thing: if you pull her away before she's really done nursing, you could alter your milk supply and in some cases this means early weaning. (This is where you hear moms say their milk "dried up." It doesn't really "dry up", you're just not producing as much because the baby stopped taking as much...you can always nurse more frequently to re-up the supply if needed but most moms don't know that and assume they have no more milk.)
If you have any questions about what I wrote, feel free to send me a message. I'm a bit sleep-deprived from having a coughing kid the past couple of nights so I might not be making much sense!
Oh, and one more quick thing. My daughter is four and has had no problems with her teeth. When she's old enough, leave a sippy cup with water ONLY (NO JUICE) in her bed which will rinse out her mouth, and also brush her teeth (or wipe out mouth with a cloth) first thing in the morning.