Is H+ (in an aqueous solution) = H3O+?
When dealing with an aqueous solution, you are correct that the $\ce{H+}$ ion is equivalent to $\ce{H3O+}$ for all intents and purposes. Due to the abundance of water in solution, molecules of $\ce{H2O}$ will readily pick up the hydrogen ions, meaning that most of the $\ce{H+}$ in an aqueous solution is actually of the form $\ce{H3O+}$. While this is the case, using $\ce{H+}$ over $\ce{H3O+}$ is more correct in general contexts. If you are using something like ethanol ($\ce{CH3CH2OH}$) as your solvent, then instead of having hydronium ions, you would have $\ce{CH3CH2OH2+}$ as your measure of concentration of acid.

As for your other questions, the $\ce{H+}$ would be considered an ion rather than an atom. This is because it is a charged species. Either $\ce{H+}$ can be referred to as a proton or as a hydrogen ion, since $\ce{H-}$ is too unstable a species to be formed. While some of your $\ce{H+}$ in solution might be deuterium or tritium, most people will understand what you are talking about if you just call the hydrogen ion a proton.