I would say the living apart thing will make your application more difficult, but It would not automatically disqualify you. As you stated here, it seems a legitmate reason and you have been keeping evidence,which is good, but it will still be scrutinized closely.

As far as what address to use, you just need to pick one. You both need to talk and make a decision at to which you are going to call your Permanant resident, at least for the duration of this process. Preferably one which at least one of you intend to live at for the next few years. The address may dictate which ICE office you have to go to for biometrics, interivews and other requirments. You should consider the convenience of which site would be easier to get to for appointments.