P<5% means there is a 1 in 20 chance the findings of a study are false; but,does that also indicate that there is a probability the findings are off by 5%?

That would be a 5% 'margin of error(MoE)'.

Say a study showed a 1 per 100 increased death rate;but, a 5% MoE would mean that anything less than a 5 per 100 rate would not be valid.

My command of statistics is sadly less than it ought to be, but I think the answer is no. For instance, if the result is binary, i.e. 1 or 0, yes or no, the result cannot be 5% off.
John has covered it in "Sorry, wrong number", but I don't have my copy to hand.

You should be careful to distinguish between P and relative risk. The nearest statistical measure to Margin of Error would be Confidence Interval. See our FAQs.