Is Mid mode the same as this tool's Average mode?
Yes, numerically identical: both compute (L+R)/2 sample by sample. Mid is the name used when the focus is Mid-Side encoding (Mid plus Side), Average is the name used when the focus is just collapsing to one channel, but the exported result is the same file.
Why does the Side channel sometimes come out nearly silent?
Because Side is the difference (L-R)/2, and when both channels are nearly identical that difference approaches zero. A centered voiceover or a mono-compatible studio recording has correlation near +1, so the Side channel of those tracks is naturally low, that is not a defect in the file.
How can I tell if a song will sound bad in mono before actually converting it?
Look at the compatibility verdict instead of trusting a single average correlation number: it merges contiguous windows below -0.1 correlation into highlighted regions, so a 15-second problem hidden inside a 5-minute track shows up isolated, even when the whole-file average would look safe.