My iPhone 3GS Won't Connect to iTunes? - 1

My mum bought an iPhone 4S I think last year and gave her iPhone 3GS to a friend. He returned it to us for he was experiencing problems with it - it would always be turning off randomly. We all believed the battery's time was up, for a few days after he gave it back to us, it would not turn on at all (held down the lock button and the Apple logo appeared for three to five minutes and then the screen would dim one third of the screen and then another third as the first section dimmed more, and so forth, still showing the Apple logo).
Recently, my iPhone 3GS--not my mum's--has only been charging using my Bose iPod dock (it's jailbroken, so it's not so unusual) and it gave me the idea to try my mum's iPhone 3GS to see if it works. Now, charging it for the first time in months (I first tried using a normal charger in a power point before trying the dock and it did not work), the dock succeeds to turn on the device and also charge it. If I leave it on the dock, charging for a minute, I have a bit of time to plug it in to a charger connected to a computer or wall. It will charge and stay turned on. I had left it on the dock overnight last night and when I took it off today, the iPhone was working. I remember when I tried using it when it was returned to us, a trigger was playing music. I played music and it continued to stay on. I then went to take a photo of something and the device finally turned itself off. I remember when the phone was Mum's current phone, in early 2012, I went to take a photo and it turned off in the very same way it did today, only the device worked when I turned it back on again.
It seems to only be the camera that makes it turn off, could anyone tell me what that means?

I really don't want to do this, but I read that restoring it to factory settings and then restoring the backup will work. But there's only one problem with performing that task: the device will not connect to iTunes. I have an unupdated version so I'm updating it now as I type this. I will plug in the phone afterwards and see if it connects. In the meantime, is there any other way to fix any of my problems?
Your help is, as always, appreciated!
Apologies for writing a book, and thank you for taking your time reading it.

- G

Added (1). - I have tried taking photos while the phone is on the dock and while charging with a cable charger and it works. It just doesn't work when it is not on charge. It does however, work taking a photo from Kik.
- The cable charger can now turn the device on, after I--using the capture feature from the Kik app--took a photo on the dock, then on a cable charger and then without a charger.
- Now, the phone will no longer connect to Wi-Fi when it is off charge

Added (2). I meant anything I could do that does not require going anywhere. That is the last thing to do. If there's anything else that does not require my mum to take me to a store, I would like to try those things first.

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