What force particle mediates electric fields and magnetic fields?
Photons, as you have suggested.
1) Wouldn't that mean that a charged particle (e.g. an electron or even a polarized H2O molecule) would constantly be losing energy from sending out photons?
You must describe this process in a quantum field theory. Virtual photons emitted by charged particles are reabsorbed in a time consistent with uncertainty principle. Hence over some finite amount of time, energy is conserved.
2) Wouldn't that mean that an electric field is inseparable from a magnetic field, as photons have both - and that one can't have one without the other?
You can already show this in classical electromagnetism - see Maxwell's equations.
3) Would it be possible, then, to determine the wavelength of magnetic-field-mediating photons? If so, what is the wavelength - is it random or constant?
The wavelength of a photon is related to it's energy, which is again related to the uncertainty principle. The longer the time borrowed from the vacuum, the lower the energy of the photon, so it has a longer wavelength. Hence the wavelength of virtual photons at large distances from the EM source is much longer than at short distances.
4) How can a photon (which has momentum) from one electrically charged particle to an oppositely charged particle cause these particles to be pulled toward each other - or how can a magnetic field cause an electrically charged moving particle to experience a force perpendicular to the source of the magnetic field if a particle with a non-zero mass moving between the two is the mediator of that force?
This become less intuitive depending on your background. Richard Feynman introduced a trick which offers a way to imagine the process. Imagine the photon is emitted between opposite-charge particles in the future and travels 'backward in time'- therefore its momentum minus minus what it really is. This is explained in good detail here.
If "virtual photons" are involved, please explain why they work differently from regular photons
Unlike 'real' photons (which have transverse polarisation), virtual photons have both transverse and longitudinal polarisations. The energy momentum four vector of the virtual photons, and generally all virtual particles, is not necessarily 0: virtual particles are off mass shell. This means that virtual photons may have non-zero mass - which means that they also have a longitudinal polarisation state. It is important to consider the extra polarisation in your calculations.