I would say no. It's actually possible for your computer to run fine with no battery, so I doubt this is the problem.
The POST code sounds like it's doing a quick error check on your RAM - performing calculations on it to check for errors.
Have a look at something like Memtest 86 and give it a good workout with that.
The screen issue could simply be needing a good degauss (I imagine it is a CRT rather than an LCD monitor?) which the switch off and restart will have provided. If it happens again, degauss it manually (usually using a switch on the front of your monitor) and let us know if that fixes it.
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