We report on the optical properties of single InAs/GaAs quantum dots emitting near the telecommunication O-band, probed via Coulomb blockade and non-resonant photoluminescence spectroscopy, in the presence of external electric and magnetic fields. We extract the physical properties of the electron and hole wavefunctions, including the confinement energies, interaction energies, wavefunction lengths, and $g$-factors. For excitons, we measure the permanent dipole moment, polarizability, diamagnetic coefficient, and Zeeman splitting. The carriers are determined to be in the strong confinement regime. Large range electric field tunability, up to 7 meV, is demonstrated for excitons. We observe a large reduction, up to one order of magnitude, in the diamagnetic coefficient when rotating the magnetic field from Faraday to Voigt geometry due to the unique dot morphology. The complete spectroscopic characterization of the fundamental properties of long-wavelength dot-in-a-well structures provides insight for the applicability of quantum technologies based on quantum dots emitting at telecom wavelengths.