P.P.
I have two boys, a 3.5 year old and a 15 month old. Neither of them were good sleepers when they were babies. I used the same routine for both of them - "rock-a-bye" time in the rocking chair with a bottle, snuggling with a book and then into the crib, usually sound asleep. Sometimes it worked and they were asleep in 10 minutes with no problem. Other times I spent an hour trying.
Some babies sleep perfectly at birth. If yours doesn't, that doesn't make you a bad parent. Some parents let their babies cry themselves to sleep every time. That's fine for them but I could never do that (with the exception of the one time I bent to the advice of others which led to puke all over the baby, the crib and an additional 30 minutes of cleanup and trying to get the little bugger to sleep.) So deciding not to let your baby cry herself to sleep also does not make you a bad parent.
Teething was bad for my kids and I kept a supply of baby anbesol at all times, along with some tylenol when it was really bad. Think about how much it must hurt to have a tooth break through your gum and now imagine how upsetting it must be not to understand why your mouth hurts so bad. Is it any wonder that a baby may not want to lay down alone with a painful mouth so they cry and want you to hold them instead.
My only advice is to hang in there. Accept that you are doing the best you can and realize that this shall pass. The teeth will all come in. Your baby will start to walk around and run and be so exhausted at night that she'll sleep. Sleeping is natural. Humans have to sleep. What makes it difficult with a baby is there is so much pressure to do everything right when you, as the parent, are so tired and exhausted yourself and you aren't sure if you are doing things right.
One last thing, your daughter is going to grow up so fast and reach an age where she isn't going to want to snuggle with you at night before she goes to sleep. Enjoy every moment you can with your baby, even if that means the dishes don't get done or the clothes don't get folded that night.