As @ELNJ answer pointed out, fully ionized the atoms at the star surface are not. It is not hot enough. Star cores are another case, but we usually don't see them. There, both pressure and temperature make impossible the existence of the usual atoms.
Atoms and molecules usually emit their characteristic wavelengths because of the electrons' energy levels...
Yes, they do. They also absorb their characteristic wavelengths, given chance.
In stars, most spectral lines we see are absorbtion - a characteristic wavelength missing (or weakened) on a background of more or less flat area. Stars with strong emission lines are rare.
Do the completely ionized ('naked') nuclei absorb and/or emit EM radiation?
If so, at what type of wavelength? Mostly visible, IR and UV like the rest of the star?
They do. They (mostly electrons, nuclei are neglected in simple calculations) scatter photons, so they emit a thermal radiation. This is the flat-ish thermal background where we see the absorbtion lines overlayed.
Nuclei have their own excited levels w/ characteristic lines in the gamma-ray spectrum, but it is not hot enough at the star surface to excite these.
Here you have an enlarged visible spectrum of the Sun:
(credit https://scied.ucar.edu/image/sun-spectrum )
Each of these black lines can be traced to some chemical element in some ionization state.