I would say that this is a good way to word it - it sounds fancier than saying "as many as he desired". In English we often eliminate pronouns if the context is clear - it makes the writing sound nicer.

I think it is confusing, because I don't know if you mean, "as many as the king desired to participate" or "anyone who desired to participate." ( who is doing the "desiring", the king or the people invited?) I think you mean the latter, so I would write, "... The king decided to have a contest. He invited all his subjects to compete to see who could travel the highway the fastest." ("Subjects" here meaning "the members of his kingdom.") If you mean the first and the king is the one desiring, "... a contest. He invited a select number of his subjects to compete ..." Or "... he invited those whom he desired to participate to compete ...".

Likewise. It means that he invited all the people who wanted to participate.

I agree with Rick, it's not as many people as *he* desired, but as many of the people who desired to participate to do so.

I will answer in a comment instead of the answer block, because, even as a native speaker, that is a tough question. I think that what you wrote is not bad. But if you change it, I think I would change it to "he invited as many as desired to, to participate."