France was not a major imperial power at the time. It was much smaller than today, lacking Savoy and much of Burgundy for start, with Normandy and many other areas only nominally part of it and technically under control of English king (who was just a duke in France, but that changed only a very little on the battlefield).

Crusades in middle east started as an attempt of Eastern Roman empire (although they just called it Roman empire / Basileia Romaion) to recover from recent advances of Muslim invaders in Anatolia (modern Turkey). But turned into an overwhelmingly religious effort in the west. The first crusade especially was largely ill organized and chaotic affair. Where on one end of the spectrum you had nobles arriving with somewhat well equipped forces and idea of what to do, and on the other you had pilgrims, with whatever they just picked up in their hands and not answering commands of anyone, but their priest.

The economic side of things came into play after the process started and gradually became dominant. But it didn't start like it.

Finally. Interest of France in Mongols can be easily explained precisely by the influence crusades had on French and other Christian elites in Europe. The initial victory of 1st Crusade was followed by a series of setbacks. Muslims gradually begun to push crusaders out, the fact that crusaders started to fight amongst themselves helped a lot.

And then mongols arrived, almost from nowhere, crushed one of most powerful Muslim states at the time, and didn't stop there. It did seem like an immense opportunity, and in a way it was. If French, or someone else in Christendom, could convince khans that some form of cooperation is possible, or even better, if Mongols converted to Christianity, there would be a decent chance to not only save Jerusalem, but to move on to Egypt (still majority Christian).

> under control of English king (who was just a duke in France,

I thought the folks in Normandy were just Nordic people who moved there and later to England