On New York Results template, can I add Contact and Schedule a showing on listing?

Yes! Now you can add those buttons, just add the code below in your Results Category in the Custom CSS panel:

div#IDX-toplinks { display: flex !important; flex-flow: row nowrap; padding: 10px 0 0 0; }
div#IDX-toplinks > a { width: auto; flex: 1 0 auto; }

Then you will have buttons like this: