Must someone first believe before being Baptized?
No. How can a newborn be said to "believe"?
In the case that someone does not truly believe before Baptism, is the Baptism still valid?
The answer to your first question will give you a clue to the answer to your second.
If a baby does not truly believe, his Baptism is valid, even if he later rejects his Lord and Saviour. Baptism is a sacrament which is done once and for all time. It cannot been repeated, nor can it be undone. It marks the soul with an indelible mark.
Your question has a protestant ring to it. Are you protestant?
There's no need to be insinuating and judgmental here. It's an honest question,
and one that deserves a patient answer.
If the person receiving baptism is of the age of reason (which could be a lot
less than 7, by the way!) and does not believe, and deliberately harbors a
mental reservation, it is quite possible that the baptism would be a sacrilege,
but nonetheless valid. It is possible to be baptized and then fall again into
mortal sin very quickly, having only endured the ritual for the sake of some
temporal gain, or some other reason. It's hard to imagine a child thinking so,
but is it impossible? This is also why it's so important for the godparents to
be good, upstanding Catholics, because they acquire a powerful spiritual effect
over the faith of the baptized, and if, perchance, a godparent were to start
practicing witchcraft, he or she would be able to introduce demonic influence
on their godchild even to the point of making demonic possession possible.
Note: the natural parents already have a most serious spiritual influence over
their own children, but the godparents acquire it upon the child's baptism.
It seems to me that for a baptism to be invalid (and it is possible for a baptism
to be invalid for many reasons) because of the lack of faith of the recipient, it
would have to be a most grievous and willful state of unfaith, for the sacrament
itself comes with powerful graces. However, the priest should never presume
that the grace of the sacrament is able to overcome the willful denial of a
recipient. The human will is a most powerful obstacle to faith.