"Free Corps, Free Legions " Topic

What was the reasoning behind forming the various Free Corps, Free Legions etc. ? Why not call them simply brigade, division, and so on ?

With the exception of the King's German Legion, and one or two Austrian legions, were they always poor quality ?

They were "free" in the sense that they were outside the established forces of the using nation. They were often made up of foreign personnel, including the officers. Some, like Lutzow's Freikorps, were privately raised and placed at the disposition of the nation. The KGL was, in effect, the Hanoverian Army-in-Exile.

And many were of good quality. Lutzow's certainly was. The Legion Portugaise acquitted itself well in Russia. The Russo-German Legion, Like Lutzow's, was good enough to be incorporated into the Prussian Army in 1815.

A key component was as raker stated. Most of these were raised when the official Prussian government was allied with France. They were financed by private citizens who got weapons and uniforms from various sources.
As noted the term Legion was used when it was a combined arms organization. There was also aStreifkorps though my spelling may be off. hellwig was the commander.
The Portuguese Legion, as noted had a strong history and plenty of men in the unit. They too had all three branches.
I am not sure about the French Dalmatia or Croatian Legion having cavalry or artillery. Maybe the artillery was assigned to forts.

There was a regiment of Croatian Hussars.

David Nash listed around a half dozen units the Prussians formed with permission of the King. Schill brother of the killed in a uprising had a Hussar unit,

The German term "frei" in the context of these unit names is sort of a short-hand for "freiwilliger," meaning literally freewill, in the sense of volunteer. Quite a few freicorps units were raised in Prussia during the desperate days of the Seven Years War, when the country was threatened with annihilation by numerous enemies. At that time, most were considered inferior to regular units, but useful for skirmishing and irregular warfare.

As others have indicated, during the Napoleonic era, the quality varied depending on leadership and discipline. I will note that although termed a legion, because it consisted of all arms, the King's German Legion was a body of regular troops forming a part of the British Army and subject to its rigorous discipline.