What to do with gap between cut baseboard/shoe moulding and new wider door trim
I'd use a non-shrinking wood putty. Caulk and white glue shrink badly, meaning you'll have to apply it multiple times. 

Mix up some powdered putty and use a putty knife to shape it to your molding. Let it fully cure and skim it again if necessary. Sand and paint. 

Next time, set your door in place (ideally you'd set the door without casing attached and then set the casing), trace the new casing edge, and remove the door. Score the trim with a knife to create a clean, sharp line, then make your cut outside that line.