Do we have any idea what percentage of Europeans could read a Bible before it was translated out of Latin? How thorough was a lay person's understanding of Catholicism if they couldn't understand/read Latin, and how did they accomplish that?

To answer the first part of your question: I've never seen a percentage breakdown, and I think it would be very hard to justify the validity of such a statistic.

The second part is more straightforward. Until the Wyclif bible appeared sometime in the early 1380s, the elements of the English populace (those not privately educated by tutors in rich households, or educated at grammar schools/universities) that could read the bible themselves would have been very small indeed.

An understanding of the Catholic faith would have been obtained through attending sermons. My own area of reading on this predominantly concerns England, Ireland, Germany and Bohemia of the late 14th and early 15th centuries; at this time each were areas of agitation for the bible to be translated into the vernacular. 'Wycliffites', 'Waldensians' and 'Hussites' respectively in each of those countries were the names attributed to groups were leading those developments, giving you a clue as to each of their theological origins (real or imagined).

How thorough the average lay-person's understanding of the bible is perhaps best understood through the control of sermons. Most scholars in the field of sermon studies accept that in the late middle ages, sermons preached to a clerical audience were in Latin, and those preached to the laity were in the vernacular, as it would not make sense to preach to an audience in a language they could not understand. This view should be nuanced by acknowledging the limitation of 'traditional' vernacular sermons elsewhere in Europe in this period. Jean-Claude Schmitt advances the idea that elsewhere in Europe vernacular sermons provided the laity with a deliberately simplistic and unchallenging doctrine, that often extended only to the teaching of prayers such as the Pater Noster. Schmitt suggests that the church would use the vernacular to transmit to the laity only conservative points of doctrine, ostensibly to protect them from heresy and superstition, but in fact serving to preserve the clerical monopoly on theological learning.

Unfortunately, scholars can only wonder as to the received understanding of Catholicism in illiterate communities that were ultimately unable to record their experiences of coming to grips with learning their faith.

Sources: C. Muessig, 'Sermon, preacher and society in the middle ages' in Journal of Medieval History, 28 (2002), p. 78.

K. Walsh, 'Wyclif's Legacy in Central Europe in the Late Fourteenth and Early Fifteenth Centuries', in Studies from Ockham to Wyclif, eds. A. Hudson & M. Wilks. Studies in Church History. Subsidia, 5: (New York: Basil Blackwell, 1987), pp. 397-398.