Running the spindle at the lowest speed is really only OK if its very slow, say less than 100 rpm. In any event the speed must not be fast enough to cause any bounce.
Whilst your plunger type indicator is not ideal you could probably just get the tip of it to bear on the on the inner surface at the mouth of the spindle taper bore.
Regarding the 'side by side' play you mentioned, its not really clear what you mean. I take it to be that you are turning the spindle slightly (CW and ACW) and seeing play in the driving splines? (if so, nothing to worry about).
If its 'side to side' movement and its not play in the bearings then it could be the quill is loose fit in the head casting.
Ian P