The main problem is that you don't know what you are doing, but think you can manage. Its all very well checking and metering things but if you do not know what you should be seeing, you can only think it is okay.
You have a 12volt battery that is only showing just over 4volts, the system is putting out 14 volts to charge a flat battery, in your words its only there if the mains fails. But ask yourself this if the mains has not failed and the battery is only showing 4 volts then something is very wrong. Now if the battery has been knackered for a long time and the system has been trying to charge this and not succeeding, things have got very hot and the panel is no longer operating to the specification it was designed for yes it will charge a battery if there has been a power failure, but it was never designed to charge a flat battery continuously. So things get hot and deteriorate etc etc and in the worst case scenario they get so hot they burst into flames, simply changing the battery may not matter as the long term damage has already been done.
Yes a faulty battery will also cause a budget panel to do strange things and not operate as expected.
If you simply replace the battery, would your house insurance cover you if the panel overheated and caused a fire..........................I doubt it.
The best advice is to update the panel and new battery. not an expensive task for peace of mind.