A weak battery isn't going to affect how the engine runs very much unless there's something significantly wrong with it. The alternator is what you need to look at if you think that's the issue.
BHG's can cause many different symptoms depending on where it's blown and how big the leaks are.
Did you read the old spark plugs before disposing of them? If not, do you still have them? If the problem is in or before the combustion chamber they'll give you a very good idea what problems you may be facing and in which cylinders.
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