Why did Cessna use a new design for the C162, instead of updating the C150 design?
I think there are a couple reasons: 1) Weight, and 2) the Type Certificate.

For #1, the original C-150 is roughly 100 lbs too heavy (in empty weight).  That's a bit over a 10% reduction, which is huge in an all-aluminum design.  Note that later models need even more weight removed (C-152 included).

For #2, the C-150 is a CAR Part 3 type certificated design.  To sell it as a LSA would almost certainly require a re-designation so as not to conflict with the original type certificate.  This is primarily a paperwork thing, but nobody had done it before (even now, none of the "grandfathered-in" type-certificated designs are still built under their type certificate or by the original manufacturer).  The C-152 is an add-on to the C-150's type certificate, so the same problem applies to it.

As far as why the C-162 failed, I think Carlo's answer covers most of the reasons pretty well (ignoring the "GA is dead" rant), though I'd like to add that Cessna did a terrible job with designing the C-162 and took so long to "get it right" that they didn't start deliveries (of an arguably inferior product) until right after the 2008 economic downturn.