Please excuse my taking several days to reply.
1.
Inductive saturation current is the plateau at which you get inductive action below that, while above that there is no further building of flux fields, but there is just DC resistance.
Your converter may operate the coil so it reaches this point many times a second.
2.
Safe current rating applies to long term use. You can exceed this rating if your duty cycle is not too great. In other words if you don't heat up the coil too much.
You may even run it up to and above maximum inductive saturation current, as long as the duty cycle is not too great.
3.
Traditional advice says to rate components for twice whatever you expect them to be subjected to. To power a 1A load you should use a coil rated for 2A safe limit.
Note that a switched-coil converter may have coil current swinging between .5 A and 1.5 amp per cycle.
Both of the coils you are considering look satisfactory.