Why has the oil separated?
Because there is no palm oil or emulsifier in it, which is a good sign. Stir it back in and refrigerate afterwards to keep it mixed.
Which nut butters have no added sugar?
Most of this range. Meridian and Carley's in particular are nuts and sometimes salt. The ingredient list is the quickest check.
Are they made in a nut-free facility?
No - these are nut products. If you are shopping for someone with a nut allergy, this is not the category, and cross-contamination warnings are on every pack.
How long do they keep once opened?
The jar gives the producer's figure, usually a few weeks to a couple of months. Refrigerating extends it and stops the oil separating again.