Over tightening a chain will make it wear faster, and wear your sprocket and chainring teeth too, though none of this should be noticable within a week or two.
Track nuts normally leave an impression in the paint with their serated bits when well tightened, so it should be quite easy to see if your axle has moved (unless you have stainless/chromed dropouts - in which case stick a bit of paint on.)
When you say 'It is more that I can feel about 5mm movement in my pedalling before drive' are you quite sure that isn't the sprocket tightening up? Double check your sprocket and lockring.
Over tightening a chain will make it wear faster, and wear your sprocket and chainring teeth too, though none of this should be noticable within a week or two.
Track nuts normally leave an impression in the paint with their serated bits when well tightened, so it should be quite easy to see if your axle has moved (unless you have stainless/chromed dropouts - in which case stick a bit of paint on.)
When you say 'It is more that I can feel about 5mm movement in my pedalling before drive' are you quite sure that isn't the sprocket tightening up? Double check your sprocket and lockring.