I've been having terrible problems with battery life on two iPhones 5s's. They get warm too.
Both have had multiple battery replacements with different companies (warranty mainly) to no effect whatsoever.
I'm convinced, as they've both been fully reset to "factory" (and still behave the same), that there is an internal problem with battery management and/or CPU.
As a result I've got very good at backups/restores with iTunes, and yes they really do work. If you use the encrypted option it saves (and restores) all the logons/passwords too.
One of the iPhones, before it starting misbehaving, would last up to 100 hours of, admittedly very light, use on one charge.
Afterwards it would discharge totally in 7 or 8 hours!
The other one was not quite so extreme but it would run down at typically 7% or 8% per hour so wouldn't last a day.
Just to be totally clear, these would be with new batteries, fully charged with a genuine Apple charger/cable and factory reset with no iCloud account, no SIM (removed after setup), and no wifi! So doing nothing at all. With a SIM in and set up normally the battery ran down at almost exactly the same rate.
I've read all sorts of stuff online about background processes, iCloud, etc. etc. but every single one has been eliminated, I'm convinced it's hardware/firmware.
Currently I don't have an iPhone as BOTH are away being "repaired" again!
Backing the iPad up to iTunes on a PC backs up all data and can restore it perfectly. I did this when transferring from my iPad 3 to my iPad Air 2 (5).
Very true but I think it was noted above that the OP doesn't have access to a suitable PC.