How to fix gap between floor and hardwood
+3
The open case mouldings were cut much higher than 3/4". Has anyone fixed an issue like this and if yes, how? Shoe moulding seems odd in an open case door frame.
Just use molding.
It looks like all of your baseboards were removed. Quarter-round moulding isn't going to be sufficient to finish off those gaps. For a truly finished look, apply baseboards and then trim them out around their base with quarter round.
Puddy and then shoe molding !
Oops - posted that answer too soon, wanted to include this informational link with photos: https://www.thespruce.com/mdf-for-molding-and-trim-1822780 The baseboards can be as simple or as elaborate as you want, depends on how much you want to spend and how much time you want to spend installing and trimming out.
I would do one of two things. You could replace the one piece of molding that we can see plainly here, that would stop an eye from catching any imperfection right off the bat. Then for the flat piece on the side I would also cut a small piece of wood and slip it into the gap space with some glue-doesn't matter too much what type but must be able to adhere to glue-and then use some drywall or latex putty preferable you might have around. Light sand that and paint the two pieces of molding. You won't ever see it. The other option is to remove the side piece and replace that. Door trim wood is pretty available at stores like home depot.
I would just add moulding.