@Greenfield
Hello there, it does indeed sounds like your A4000 is a little poorly.
Sadly I am sure to suspect that your on-board battery has leaked - as such it has caused damage to your A4000 Motherboard.
With the symptoms you have described I am sure that a couple of IC's will need to be replaced as well as some capacitors - not to mention the removal of the battery and motherboard clean up.
At this juncture I would suggest you open your A4000 and remove (gently) the Battery - from this I would use some lemon juice on the motherboard - be liberal - and let it rest for about 10 minutes - again gently - with a toothbrush have at the motherboard (primary area will be the side of the Battery - but get as much as you can.
Once you have done this I would recommend IPA (Isopropyl Alcohol) and be liberal when you apply this to the motherboard - gently work this in and then give it 30 minutes dry time.
Make sure the rest of the board is dry.
Now, this WONT repair the problems but it will ensure that it doesn't get any worse. The motherboard will need some serious repair at this stage - it will require SMD reworking - soldering skills.
do you know anyone local in your area ?
If not you could ask here, there are also members whom offer repair services on
www.amibay.comI hope that helps.