Why LM358D and LM358ADT differ in price a lot?
To see the difference between the parts, have a look at the electrical characteristics table in the datasheet:



The 'A' types have lower specs for input offset voltage and current. This probably means they are cherry-picked parts from the same production run. 

The 'A' types also have lower specs for offset drift over temperature. This might mean they are tested over temperature (where the others might be guaranteed by design), which adds significant costs in production.

But in fact I don't see any 5x difference in price. LM358ADT are $0.51 at qty 1, while LM358DT are $0.57. 

As for why the 'A' parts are actually cheaper than the others, it's probably just supply and demand. For low volume designs, Mouser customers are probably buying more of the 'A' parts, allowing Mouser to buy from ST in higher volume and get a better price. Notice that if you buy a full reel (2500 pieces) or more, the 'A' parts are priced higher.

The parts priced above $2 each appear to be the parts packaged in tubes rather than tape &amp; reel (or cut tape for small quantities).