No, the missing Cat won't cause the running issues, but it is a $2,500 Federal Fine for not having it, so don't take it to a shop for any work, they are required to report it or install one.
Yes without the cleaning of the exhaust by the Cat the rear O2 sensor, if it is still there, will turn on the CEL, if it has been removed then the CEL will come on because of that.
With engine warmed up and idling, unplug the IAC Valve, idle should drop to about 500rpms, or engine should stall, either is fine, it means no vacuum leak.
If idle doesn't drop below 600 then you have an air/vacuum leak.
PVC valve hose is a common leak on Rangers.
Air Plenum tube from MAF to intake also needs to be air tight, that is a "vacuum" too.