Will using a 1:1 balun at the feedpoint of a center-fed dipole when using coax result in no common mode current on the outside of the coax ?
Depends on how much current you mean by no current.
But in general, unless the dipole is actually perfectly symmetrical, in a perfectly symmetric environment, above a symmetric ground, with the feed line at exactly 90 degrees all the way into the far field, and/or with the radio equipment and their ground connections also exactly 90 degrees perpendicular to the dipole, any of those could cause asymmetry in the RF field around the antenna, which could then couple into the outer coax (and any other nearby conductive objects).
It might be possible to reduce that coupled current into the coax shield by positioning chokes which can increase the impedance to asymmetric RF fields, if any.
But odds are that if you were to place a short receive antenna anywhere along your coax, exactly parallel to it, you would pick up the transmitted signal. And if that short receive antenna is picking up RF, so is the much longer coax.
Yes. If the balun is properly designed, there will be little to no common mode current on the outside of the shield.
It is almost always good practice to use an effective choke balun at the feedpoint of any center-fed balanced dipole. The exception would be if some vertical radiation was helpful and the coax was not picking up RFI.
Hotpaw2 makes some excellent points about being symmetrical in his answer. I doubt whether a "small" amount of asymmetry would cause a significant amount of common-mode, but that would have to be determined by either experimenting or modeling.
Regardless of the amount of asymmetry, the choke balun would help; however, at some point it would become necessary to add a choke farther away from the feedpoint.
Here are some past questions and answers I found searching for Balun center dipole.