We've got a few different questions here, and a lot of the answers vary depending on time and place, but I'll do my best. Some caveats: the best book on this, Julia Barrow's The Clergy in the Medieval World (Cambridge, 2015), is something I havn't read in depth yet, and don't have with me at the moment, but I'll double check when I'm back in uni on Monday if there's any points in particular you'd like further information on. Second caveat is that I'll limit my answers to roughly the period between the Carolingian Empire and the Fourth Lateran Council (1215), because these both heralded some big changes for the clergy, and the period between is the one I know best.

First point is, although you say

It didn't have the option of simply recruiting members from birth like the second estate, after all.

..actually, it did, or it did at least some of the time. For a long time, the priesthood, especially lower clergy who would have ministered to the ordinary people, was a hereditary profession, with the position of local priest (later parish priest, after the introduction of the parochial system) passing from father to son. This was because clerical celibacy was not a requirement for the secular clergy (i.e. ordinary priests, as opposed to monks/canons/(later) friars etc.) until the mid-eleventh century. The ban on clerical marriage which was pushed by the reforming popes from c.1050 onwards faced serious and sustained opposition from bishops and priests across Europe for decades, which formal, public opposition still visible in the early 1100s. I think it's safe to say that the practice of clerical 'marriage' went on informally throughout the middle ages, although once they were officially banned, it because much more difficult to pass the office from father to son.

One of the reasons for this is because, to answer the second part of your question, no, not anybody could join the clergy. I'm not super clear on the requirements, but there were certain things which could rule you out. Illegitimacy was one of them, hence the difficulty that once priests could no longer be married, any children they had, on whom they hoped to pass on the family trade, were barred from sacred office. Legal freedom (i.e. not being a serf or slave) was another requirement. I can't really remember the rest - I'll check Barrow's book on Monday. I had thought physical dis-figuration or disability was one other bar, but I've just remembered Hermann of Reichnau, a paraplegic monk who wrote an important chronicle in an Alpine monastery during the reign of Henry IV, so I may be wrong on that count.

There were other barriers to entry, however, even if they weren't always particularly high. The main one was that to be a priest you needed training. There's debate to the level of literacy among the clergy, but at the very least you needed to learn by heart the psalms and all the major liturgies of the church. Of you were a monk or had any aspirations at being anything other than a country priest, you would have had to learn to read (by which I mean read Latin). And, of course, these things cost money, which would have had to be paid by your family. This doesn't mean that all clerics were drawn from the nobility, just that they are unlikely to have been drawn from the poorest of society (i.e. landless labourers, farmhands etc.).

Could a peasant become a member of the elite by joining the clergy?

Yes. But it was incredibly rare. It's also very difficult to determine how low-born these cases really were, because we usually know only that their enemies taunted them for being peasants, but medieval writers were incredibly snobby: to them being the 3rd son of the Count of Neverheardofitsville was practically the same as being a peasant, even if that wasn't the reality.

Hildebrand, the future Gregory VII, is the best example of this uncertainty: he is very often claimed to be of humble origins, the son of a Tuscan carpenter, but there are also suggestions that he was related to a previous pope, Gregory VI, and that he had family ties with some pretty important Roman monasteries. We do know that peasants could make it to the top. The one case in which it's universally accepted is that of Pope Silvester III, better known as Gerbert of Aurilliac (b. c.940), who rose from his early life as the son of a peasant from the mountains of the Auvergne in central France, to holding important ecclesiastical office in Spain, Germany and France, tutoring the Emperor's son, eventually being elected to the papacy (in 999) and generally being regarded as the most influential intellectual of the tenth century. But Gerbert was so unusual that future churchmen speculated that he could have been in league with the devil, his rise was so spectacular and unusual.

But on the whole, the higher clergy (bishops, abbots, popes) were composed of men from the noble strata of society, often the younger sons of kings, dukes and counts, or the nephews/cousins of older prelates. Many appointments to church office were filled through patronage and influence: even after the reforming movement tried to regularise appointments from the 1050s onwards, politics played a huge part. For example, I'm currently studying two French dioceses in the eleventh century, and I can't find a single non-noble bishop in either of them, while the most powerful, influential and numerous are the sons and brothers of counts, viscounts and kings.

For more reading on specific examples:

• H.E.J. Cowdrey, Pope Gregory VII, 1073-1085 (Oxford, 1998)

• Pierre Riché, Gerbert d'Aurilliac (Paris, 1987) (Sorry if you don't read French, but that's the only book which focuses solely on Gerbert that I know of.)