Yes, that’s an acid base reaction between basic hydride (from BH4) and the moderately acidic glycerol. For this reason we often cool the reaction mixture in an ice bath (or dry ice bath) and use a slight excess of borohydride.

Thank you!

Reaction should be very fast, in particular in basic conditions since base will catalyze conversion of the anomeric center to the ring-opened aldehyde form (which is reduced). I’m not sure what procedure you’re referencing, but it’s generally very quick.

There are many, many preparations out there for reductive amination with NaBH4. Byproducts are generally not a problem. Is there something specific you’re seeing?

Well, probably has something to do with higher electronegativity of boron relative to aluminum plus poorer Al-H orbital overlap versus B-H overlap

Are you sure it’s NaBH4 and not something like lithium triethylborohydride (Super-Hydride) which is much more reactive? I’ve seen examples of it reducing secondary alkyl halides, but not tertiary.
e.g. https://www.sigmaaldrich.com/deepweb/assets/sigmaaldrich/marketing/global/documents/157/575/acta-vol12.pdf
see page 7 on that pdf

BH3 would be the first byproduct. I didn’t go into more detail than that. In air, BH3 will rapidly oxidize to B(OH)3

Usually one would use a slight excess (e.g. 1.1 molar equivalents)

Lactones are slightly more reactive than normal esters. That being said, I’d still use LiBH4 instead of NaBH4 for lactone reduction.