Need guidance in designing my trip

Cortina is always open. It’s a town where actual people live all year around.
If your plan is to visit those locations before your conference then the order could be
Cortina
Santa Margherita
Umbria
Abruzzo
Rome

Cortina is probably closest and most easily reachable from the Venice airport. However if you fly from Liverpool then your best option might be RyanAir to Bergamo (BGY), then from Bergamo to Cortina via train and bus. There are no non stop flights from Liverpool to Northern Italy except for BGY.
You can go to Santa Margherita by train
Umbria is a region not a city, so you need to decide where exactly you plan to visit, but you can go there by train.
Coastal Abruzzo is also reachable by train, and Rome is also reachable by train from Abruzzo.

It would make sense to fly at the end of your conference from Rome to NYC.
If you want to extend your vacation and postpone your return beyond your conference, you could do your Abruzzo trip after the conference, then return to Rome for the flight back.

Unfortunately, except for Rome, your destinations are not easily reachable from any nearby international airport.

Consider this an addition to my original post.
Cortina in early June isn’t recommended & so I seek the advice of this forum.
Per online and past RS Travel Forum posts,
Until after June 15 or 20:

The Cortina gondola isn’t running
Cortina restaurants and hotels are not all open
Hiking can be muddy or have otherwise unfavorable conditions

Therefore I seek advice of the most efficient travel modes and order of stops in which Cortina is at the end of my journey.

Thanks

There could be some restaurants and some hotels that are closed, but there will be many hotels that will be open and able to accommodate you and of course there will be plenty of restaurants that will be open for local Italian tourists and local residents. Cortina is not a tiny village, it is a town of over 5,000 people so I’m sure they will have at least 3 restaurants open for your 3 nights there. Regarding the mud on the trails there is no guarantee the weather on June 15 or 20 will be sunnier than June 9, so you will need to take a chance no matter what. Since you are at a conference in Rome on June 18-21, if you want to go to Cortina after June 20, then your only alternative is to prolong your vacation and go to Cortina after your conference. In that case you may want to do Abruzzo and Umbria first before the conference, then the northern destinations later after the conference. If that is your choice, then you may want to return to NYC from Venice or from Milan, which are in the north. There are flights from Liverpool to Rome Ciampino (CIA), still with Ryanair.

A town not open? Just go to booking.com put in the town and dates. i got back at least 20 hotels and apartments in and around Cortina d'Ampezzo Italy to stay June 9th for a few nights. As for mud, that depends on the weather which is always different. -Bill