New one is 28000/- nowadays. Killer deal if you get a good one.
Remove memory card if you are using one and then check. Does it happen in burst shots only?.
Take it to any repairing shop and ask him to check it with multimeter, he will check within a minute by connecting multimeter probes with battery connector pins. Do not let him open the phone.
Depends on many factors, warranty status etc. Non warranty is usually 50-52 and warranty 60-62. ~10000/- difference between warranty and non warranty models.
Theme changing not possible. Factory reset the phone after update and also wipe cache partition.