You have to buy insurance as the owner & insured must be the same person. Insured period can exceed roadtax, not vise versa, so depending on the current road tax. Cancellation of roadtax can be done at JPJ during change of name, balance will be refunded but need to fill in form & must photocopy bank book with account details.

Yes, all this BEFORE you go JPJ, they are the last step.
Inspection is valid for 1 month, however since you mentioned the previous inspection done for the first name change and you no longer have the PUSPAKOM cert of inspection, which you will need to produce during change of ownership at JPJ, I would guess you'd have to do inspection again... Call up PUSPAKOM & JPJ to clarify.