well, If you are sure that your Z is getting hot, Im affraid that it is no good sign about your Z... Hum, with this the only last resource before send it to service is try to find another Z that could be used to compare with a battery that is for sure that is good... in that case, as I said previously, you could check a cell battery store about a replacement... about why the Z fail only for stay disconnect,I reallycould dont say anything, because is really rare, and about the service cost, I could said you noting too, because really dependes of the country where you are.... and I REALLY KNOW how hard and EXPENSIVE could be the service if you not are in Europe or USA.... and at least me, Im very clear that the Z now reaches a LOT OR OTHER countries around the world EVEN IF sharp didnt had a representative in there....
well, one last hope is that your Z could be heated by a side effect of a damaged battery, but if this is the cause, is very possible that your Z motherboard trouble could be in the charge circuit, so may be it could be because a faulty rectifier diode or a damaged capacitor, but if the factory service is so expensive that the most reasonable option is to discard your Z, you could also try to servic it with a local electronic repair service or try to sell your Z for spare (for the screen may be you could get some reaonable money that could you help with a replacement...)
Sad that I could said something more helpfull to you, but I didnt had more ideas that the previous or let your Z disconnected for a week before retest the tihings
Best regards!
