Part 1 - Context

The source material for late medieval France specifically is a little sparse, but there wasn't anything particularly special about France compared to its neighbours so I hope you don't mind me using evidence from places other than France. The reason this question has taken so long for anyone to answer is that the source material needed to answer these questions are sparse and difficult. People writing in the Middle Ages cared about kings and feats of valour; when a trebuchet is demolishing a castle wall, no source cares to record the long chain of events that brought said trebuchet to that moment. The best we’ve got in terms of sources are financial records, mostly English, which record these people being paid. Imagine trying to work out say… the life and background of a modern pilot based only on the quarterly reports of American Airlines, but where most of said reports were damaged in a fire. So it’s best to see this answer as “Here is how I think it worked based on extremely fragmentary sources that are insufficient but it’s all we’ve got so I might be wrong about some things” rather than “Here’s how it worked, bask in my genius.”

An important thing to note about this question is that there were generally two groups of people who would be part of a trebuchet crew. The first were just rank and file soldiers, or local labourers, who were told to build and operate the machine. They provided the basic manual labour. The answers to your questions for this group are very simple: they volunteered or were ordered to do it, no vetting process required beyond 'can you use a hammer?'. The main downside was that your own machine might malfunction and kill/maim you, the main upside was bonus pay and avoiding the front lines, though the enemy's archers and artillery would've been doing their best to kill you.

The second group of people, who I suspect are the people you're more interested in, were specialised carpenters and engineers who designed the trebuchet, constructed the most complicated aspects of its design, supervised and directed the labourers, and sometimes aimed it using maths. They usually also had the job of repairing the wall they just knocked down after the castle or town was taken.

I also think a bit of context the logistics of western medieval trebuchets might be helpful. From the 13th century, it was increasingly common for trebuchets to be flat-pack. That is to say, the components were assembled before the campaign and then brought along to an enemy castle or town in wagons. They were then assembled by a team of carpenters and labourers. This innovation made them quicker to build, easier to transport, reusable across multiple sieges, and allowed for standardised designs that could be produced in bulk. The flat-pack trebuchet seems to have been an innovation of Richard I's engineers for the Third Crusade. Richard anticipated sieges but also knew that the Holy Land did not have sufficient forests with sufficiently strong wood to support ad hoc trebuchet and catapult construction. Richard decided to bring two massive trebuchets with him, but they needed to fit on the ships Richard was using to ferry his army to the Holy Land, which forced the innovation of a compact transport solution. By the later Middle Ages it was commonplace for armies to travel with wagons full of trebuchet parts.

This means that a trebuchet crew actually had two sides to it. There were the people who operated the machine in the field, and the people who originally built it. For the purposes of answering the question, I'm going to call that first group the 'field team' and the second the 'design team'. They would often overlap, especially for non-standard machines where the carpenters who built it might be paid to come on campaign and supervise their machine themselves. Due to the nature of the evidence, we know far more about the design team than the field team.