An image is converted from 32-bit to 8-bit. Is any information lost after converting it back to 32-bit?
I general, yes... but strictly speaking, that will depend on your image. If it was (for example) all black beforehand, nothing will be lost ;)
In PNG images, colors (and color components) are usually encoded with 1, 2, 4, 8 or 16 bit per pixel and color channel (e.g. R, G, B), so "32 bit" would probably mean "8 bit/pixel plus an alpha channel byte". If you reduce the image size by converting to a less-bit per pixel variant, less different values can be distinguished - information is lost. When you convert the image back to a higher bit-per-pixel encoding, it's impossible to reconstruct the original pixel values (but you at least will get similar ones).
There is an important exception: palette-based encoding. If there are only a few different colors (256 or less), you may encode each color with a single byte (8 bits) - this value for each pixel serves as an index in a separate table of colors (the "palette"). This palette will still have full 8 bits per color (24 bits for RGB), although the image is "converted to 8 bits per pixel". - This approach is mostly used with logos or synthetic images (which just contain a handful of different colors); it is not suited for photos.
The information is lost when you convert it to 8 bit - the remaining 24 bit go in the trash. Converting it back makes each value a 32 bit value again, but the 24 lower bits are all zero; there is nothing to put there.
Is any information lost after converting it back to 32-bit?
When you first convert to 8 bit information is lost, not hidden / compressed ... (unless 32 bit already contained information can be stored on 8 bit too).
When you convert it back to 32 bit you don't have any sources to get back that lost information.