Not an expert on France in that period, but I will attempt to answer the second part of the question from the broader perspective of comparative constitutional / institutional history.
In that era the clergy in general have, in addition to their specifically religious functions, important civil and administrative duties. There are several reasons for this:
They do not necessarily have a monopoly on literacy (the gentry, merchant classes, and even the higher artisan classes, are increasingly literate by the late Middle Ages), but they do have the leisure to devote to learning. The clergy is also set apart from the rest of society by its special rules and codes: it is not just an educated class, it's an educated corps or cadre. (The word clergy comes from the same root as the word clerk).
One of those rules was that, before the Reformation (and, in Catholic countries like France, after the Reformation), the clergy did not marry or have children: they were therefore portable (they could move without having to uproot families), available for arduous administrative work (they did not have to worry about devoting time to domestic responsibilities), and supposedly resistant to nepotism and corruption (they did not have to make money to pass on to their sons). Monks also took an oath of poverty, so in principle (although standards became lax and some Abbots lived in luxury) they were not trying to take public money for their own use.
Bishops and abbots were major landowners and often had civic or administrative functions in that capacity - as lords of manors, heads of manorial courts etc. Even lower clergy had certain (myriad) administrative functions at the parish level. They were used to judging, resolving disputes, and managing people, money, projects.
Sometimes the higher clergy had public governance functions, which might be based on charters from the Crown or other recognised rights of sovereignty: there were 'Prince Bishoprics' in several parts of Europe (e.g. Liege, in current Belgium) where the Bishop was also the ruler of a territorial state. Several bishops were numbered amongst the electors who chose the Holy Roman Emperor. Bishops often lived in palaces, held courts etc. Despite the power and centralisation of the English state, the Bishop of Durham held his own court, minted his own coins etc.
Often, bishops and abbots were included in Parliaments or meetings of the Estates of the Realm. The details vary (as did the powers, the autonomy from the Crown, and the degree of institutionalisation of these bodies), but in general, across late medieval and early modern Europe, this parliamentary (or proto-parliamentary) activity would put bishops in contact with the King, his court, and political life.
Again, before the Reformation - and in Catholic countries after it too - clergy were seen as belonging to a society (the church) which cut across national lines. Although they might serve their own king or state, they also owed another allegiance. This made the clergy (and monks) useful as go-betweens, diplomats, messengers-to-parly and so forth.
The clergy was one of the few careers open to talents. Don't get me wrong, the younger sons of aristocrats were more likely to end up on the bishop's throne than the sons of peasants - but, still, there was a way for bright, ambitious people of relatively humble origin to climb the social ladder through the church.
Although in some places it was developing, in most countries wasn't really an established 'Civil Service' as such in this period. We can still think in terms of a King governing largely through their 'Household' (retainers or personal servants of the King) rather than through 'Public Officers'. The Chapel was an important part of the Household, with responsibility not just for ecclesiastical matters, but also potentially for the distribution of patronage, the hearing of requests etc.
All of these things made reliance on senior clergy (cardinals, archbishops, bishops and abbots) natural in the late medieval and early modern State. France is not exceptional in that regard. In England, for example, the Lord Chancellor (who was really the senior official of the King, combining the roles broadly of Prime Minister and Chief Justice today) was invariably a clergyman during this period. On the other hand, don't forget that the clergy were only one part of the elite, and that while many of them might have been in high positions, other positions were filled with lay aristocrats, knights etc - who had the advantage (in principle) of being able to fight and lead armies.
Sources:
A. R. Myers (1975) 'Parliaments and Estates to 1789'.
Luise Schorn-Schutte (1998) 'The Christian Clergy in the Early Modern Holy Roman Empire: A Comparative Social Study', The Sixteenth Century Journal, Vol. 29, No. 3 (Autumn, 1998),