Blast reports E-values, but short-read mappers report mapping qualities. Are they the same thing? Can they be converted to each other? If not, why blast doesn't report mapping quality while short-read mappers do not report E-values?
The E-value and the mapping qualities are two very different things.
The E-value is "a parameter that describes the number of hits one can 'expect' to see by chance when searching a database of a particular size". More details can be found here: https://blast.ncbi.nlm.nih.gov/Blast.cgi?CMD=Web&PAGE_TYPE=BlastDocs&DOC_TYPE=FAQ#expect
The mapping quality is an attempt to estimate the probability that a given base from a given read is mapped correctly to a particular place in the reference genome. Different aligners calculate mapping qualities very differently, so there is no simple way to compare them across aligners. But in general the mapping quality will consider things like: (i) the quality of the base call; (ii) the repeat structure of the reference; (iii) the alignment algorithm; (iv) whether the read has a mapped pair; (v) anything else the author of the mapping software thought might help.
E-values and mapping qualities are, therefore, measuring two fundamentally different properties with very different uses.