Once she receives the sacrament of Baptism, is she required to convert to Catholicism for that Baptism - and therefore the marriage - to be valid in the eyes of the Church and in the eyes of God?
First off, Baptism is pretty big deal in the Catholic Church. In order to ensure that the adult is serious and understands what it entails, they must go through the Rite of Christian Initiation of Adults (RCIA), a series of classes about the beliefs of the Catholic faith, typically taking place over several months.* At the end of that period, usually at the Easter Vigil, they publicly profess that they believe all that the Catholic Church teaches, and express their desire to join the Catholic Church. That is, they are publicly converting to Catholicism. They then receive the sacrament of Baptism, along with both the sacrament of Confirmation and their first sacrament of the Eucharist.
In other words, converting to Catholicism is a prerequisite for Baptism (and the accompanying Confirmation and Eucharist). While you could certainly lie when you publicly say that you believe all that the Catholic Church teaches and then get baptized, that would make the Baptism invalid in the eyes of the Church, as well as the consequent sacrament of marriage (even if it's never "caught" by the Church).
If your girlfriend is not prepared to leave her Buddhist faith and convert to Catholicism, then she cannot become baptized.
* Some parishes allow individuals to join RCIA if they want to become educated in the Catholic faith, but do not intend to convert or be baptized. It's not uncommon for an interfaith couple to join RCIA together so that the non-Catholic better understands their spouse's faith.
I have read that a Catholic can even marry an unbaptized non-Catholic within the Church, but that such a marriage "would be valid, but not sacramental."
This is correct. The sacrament of marriage can only happen between two Catholics. However, non-sacramental marriage* between a Catholic and non-Catholic is permitted and valid if the proper procedure is followed. A Catholic wishing to marry a non-Christian** would need to request a "dispensation for disparity of cult". This process is described in the Catechism of the Catholic Church (CCC) sections 1633 through 1637, in particular 1635 (emphasis in the original):
According to the law in force in the Latin Church, a mixed marriage [i.e. two Christians where only one is Catholic] needs for liceity the express permission of ecclesiastical authority. In case of disparity of cult [i.e. marrying a non-Christian] an express dispensation from this impediment is required for the validity of the marriage. This permission or dispensation presupposes that both parties know and do not exclude the essential ends and properties of marriage; and furthermore that the Catholic party confirms the obligations, which have been made known to the non-Catholic party, of preserving his or her own faith and ensuring the baptism and education of the children in the Catholic Church.
In other words, you must:
- Get explicit permission from an ecclesiastical authority
- This is usually the office of the Bishop of your diocese
- The exact process will vary depending on your diocese, but nowadays it's typically started by talking with the priest/deacon of your parish, who then submits a form to the Bishop. Here is the form for the Diocese of Honolulu, and here is a similar form for the Diocese of Ossory.
- Be in agreement with your future wife about "the essential ends and properties of marriage", as understood by the Catholic Church
- The most basic of these ends and properties would be that it is a life-long monogamous union between a man and woman
- As a Catholic, ensure that any children of yours are baptized and educated in the teachings of the Catholic Church
If these things are followed, then you are free to marry your girlfriend even if she chooses to remain Buddhist.
* The difference between a sacramental and non-sacramental marriage is partially described in this question. If you have further questions about the difference, it's best to ask a new question on this site.
** The process of marrying a baptized Christian of a different denomination (called a "mixed marriage") is similar, but has a few differences that are best answered in other question.
If we were to marry in this fashion without her Baptism, is there a danger of living in mortal sin as a result? I assume if that were the case the marriage would not be allowed in the first place, but this is a grey area I would like to have clarified.
No, if you follow the process of being granted a dispensation for disparity of cult, you are not living in any sort of mortal sin (if you fail to get the dispensation, then you would be committing fornication, which is a mortal sin). Indeed, CCC 1637 encourages the Catholic to practice their faith in the hopes of leading their spouse to join the faith of their own volition.
In marriages with disparity of cult the Catholic spouse has a particular task: "For the unbelieving husband is consecrated through his wife, and the unbelieving wife is consecrated through her husband." It is a great joy for the Christian spouse and for the Church if this "consecration" should lead to the free conversion of the other spouse to the Christian faith. Sincere married love, the humble and patient practice of the family virtues, and perseverance in prayer can prepare the non-believing spouse to accept the grace of conversion.
So the Church, rather than believing you to be in a state of mortal sin, supports your marriage (so long as the criteria above were followed) and hopes that in the future it will lead your wife to convert to Catholicism of her own free will.