Do young field stars exist? Or are they fundamentally impossible, such that young stars can only be members of open clusters?

They can exist at least in the form on runaway stars, with very large space motions of roder 100 km/s with respect to local Population I objects. When the data are good enough, these motions are found to trace back to star-forming regions, and there is a plausible case that they were ejected in three-body encounters involving original binaries. Examples include Mu Columbae and AE Aurigae; these two may have come from the same event, since backtracking puts them both in the Orion Nebulae around 2.5 million years ago.