Usually when I get a customer complaining of a 'rubber burning' smell it turns out to be an oil leak dripping on hot exhaust. Sometimes a belt slipping over a frozen pulley (but that doesn't last very long
) or electrical insulation melting.
You'll be lucky if you don't have to pay someone to take your core - best bet is to take it to AutoZone with your receipt (to show that you bought its replacement there.)
Sudden battery failure is normally due to a broken cell connector or an internal short. If I remember correctly some Die Hards are calcium-whatever batteries and they hold a great amount of energy but when they completely discharge they can take three to four days of slow charging to completely restore. Just to be on the safe side I'd let someone check the alternator output.
Beav