Can either parent claim our daughter if we are not married but live together and both contribute to her support? Can we switch who claims her from year to year?

If you are not legally married but all of you live together as a family then the one who has the highest AGI should claim the child; that parent can file a Head of Household and get the child-related credits.   The other parent files as Single and does not enter anything on their tax return about the child.

Yes, if two parents live together unmarried, and share custody, then either parent can claim the child as a dependent (but not both) and you can vary from year to year.  The parent who is not claiming the child should delete the child completely from their return to prevent confusion, and the parent who is claiming the child should answer "No" to the question about a custody agreement, since that question only refers to a legal agreement between parents who are divorced or separated and live apart.

In general, if the parent who pays more than half the household expenses (rent, food, insurance, utilities, repairs) claims the child, that parent can also file as head of household, which has lower tax rates than filing as single.  This is not necessarily the parent who earns the most, depending on how they spend their money, it is the parent who pays more than half the household expenses.  It only needs to be $1 more than half, but if you each pay exactly half the expenses, neither parent can file as head of household because exactly half is not more than half.   The other parent would file single with no dependent.  This will usually result in the lowest overall tax or largest combined refund.   If the parent who pays less than half the expenses claims the child, then both parents file single.  

However, the only way to know for sure which filing method results in the lowest tax is to test both combinations.