By what process did the miraculous conception of Jesus occur? Is there an official position in Roman Catholic doctrine about the following: when Jesus was conceived, was it from an egg cell of Mary fertilised in a supernatural way, or did Jesus's embryo directly appear in Mary's womb?
The official Catholic doctrine on the matter is in accord with a commonly held belief among nearly all Christians that Jesus was conceived within a virgin, and that the process by which this occurred was a mystery and a miracle. The heresies of the Psilanthropists and Adoptionists are notable early exceptions, and modern criticisms of the doctrine come nearly exclusively from non-Christian sources.  

ReligiousTolerance.org says it nicely,  


  The Anglican Communion, Eastern Orthodoxy, Protestant faith groups and Roman Catholicism have taught the "virgin birth," although the term "virgin conception" would be much more accurate. This has long been one of the Christianity's foundational beliefs, along with the inerrancy of the Bible; the God's inspiration of the authors of the Bible; the atonement, resurrection, and the anticipated second coming of Jesus, etc. All of the commonly used major ancient church creeds have also mentioned the virgin birth.   


Specifically, the official doctrine of the Catholic Church is summarized in paragraph 497 of the Catechism of the Catholic Church, 


  The Gospel accounts understand the virginal conception of Jesus as a divine work that surpasses all human understanding and possibility: "That which is conceived in her is of the Holy Spirit", said the angel to Joseph about Mary his fiancee. The Church sees here the fulfillment of the divine promise given through the prophet Isaiah: "Behold, a virgin shall conceive and bear a son."


The Catechism also gives the following in the third statement of the Apostle's Creed, in reference to Christ,


  He was conceived by the power of the Holy Spirit and born of the Virgin Mary.


Since the understanding of biological organisms and human reproduction did not include cellular phenomenon until the microscope was invented (the spermatozoid was first discovered in 1677 by Anton van Leeuwenhoek and the ovum in 1827 by embryologist Kart Ernst Von Baer) it is not possible for such a detailed  scientific description of the event to have been posited until long after the canonical doctrines were established. 

Even so, possible mechanisms of the virgin birth have been discussed in a paper published in Science &amp; Christian Belief  by R. J. Berry. The abstract from that paper states, 


  The Bible describes Jesus as being born to Mary ‘by the power of the Holy
  Spirit’, implying (although not stating) that the Spirit was his father. This has been rejected by some as an unnecessary doctrine, separating Jesus from
  the rest of humankind and dependent on an intrinsically incredible miracle.
  Such an objection is wrong: some form of distinctiveness like a Virgin Birth
  is theologically required if Jesus is to be divine as well as human, and there
  are several mechanisms by which the virgin birth of a male child could
  occur. The reason for recognising these is not to suggest that God
  necessarily used any of them, but simply to point out that apparent
  scientific difficulty should not determine the acceptability of a theological
  concept.


The paper not only discusses hypothetical mechanisms of spontaneous development of male children in human females, but also gives a nice overview of the development of the doctrine in the church and a history of its opposition, mostly from outside of the Christian Church. I expect that reading that article will complement this answer to address your question more completely.