Is H3O+ Polar or Nonpolar?

Answer: H3O+ is a polar molecule due to the existence of a pair of lone pair electrons on top of the molecule causing electron-electron repulsion. This results in a "bent" structure that leads to an unequal distribution of charge within the molecule. Furthermore, the positive charge on the oxygen causes this molecule to be a cation which therefore automatically exhibits polar qualities. 

Hydronium is the actual product that forms when people imagine H+ protons within the context of an acid. Individual protons are exceedingly rare, oftentimes H3O+ is formed instead. The transfer of H+ to H2O to form hydronium is what lowers the pH within solution and thereby increases its acidity. All acids are in some sense measured based on their strength relative to hydronium because strong acids are essentially better proton donors and weak acids worse proton donors within solution. According to the international naming conventions, H3O+ is actually supposed to be referred to as oxonium although this is not commonly practiced. 

H3O+ Ball and Stick Diagram. Created with MolView.

Besides as an acid in H2O, how else does H3O+ appear?

There is a possibility for H3O+ to combine with negative ions. However, in order for these kinds of compounds to be stable, you will need to make sure your are utilizing a very strong acid. These kinds of salts also form with other combinations of hydrogen and oxygen atoms (such as H5O2). H3O+ also naturally appears in interstellar clouds as the result of reactions turning H2 to H2+ before combining with hydroxide (OH-) atoms. This is a quite interesting observation since the purpose to help ensure that space is ion-neutral.