Does the hydrogen ion actually exist?
Yes free $\ce{H+}$ ions, protons, really exist.  

Protons are constantly emanating from the sun and reaching Earth.  

The proton flux is continuously monitored by satellite.  

However, in a solution such as water, instead of bare $\ce{H+}$ ions, they are $\ce{H3O+}$ or larger ions such as $\ce{H5O2+}$ or $\ce{H9O4+}$.  

When $\ce{HCl}$ dissolves, the proton is transferred to a water molecule to form $\ce{H3O+}$.  

$\ce{HCl}$ only has 17% ionic character according to Pauling's The Nature of the Chemical Bond, so it is covalent, but reacts with water to form separate ions in solution.  

For more information see Eigen or Zundel Ion: News from Calculated and
Experimental Photoelectron Spectroscopy:


  The appearance
  of the hydronium ion was and is
  currently debated as several possible
  structures, three of them being $\ce{H3O+}$
  (the Eigen cation), $\ce{H5O2+}$ (the Zundel
  cation) and $\ce{H9O4+}$ (the strongly solvated
  Eigen cation)


and 

Both Zundel and Eigen Isomers Contribute to the IR Spectrum of the Gas-Phase $\ce{H9O4+}$ Cluster:


  The “Eigen cation”, $\ce{H3O+(H2O)3}$, is the most prevalent protonated water structure in the liquid phase