You can buy the Eurail Pass online and use it as mobile pass on the Rail Planner app. I didn’t buy one, but it should be delivered a code via email to enter in the app, it should not take so long to do this. 

Depending where you want to go, you should check if you need for this route reservations and if there are seats left for this route (Spain, trains from and to France and Eurostar/Thalys has only some seats for Eurail users and can be booked out). 

Without exact detials impossible to say. It covers 33 countries (but 1 of these is free anyway) and every country has its very own rules. BUT about any newbee seems to want to take exactly the hard to get superfast long-distance trains so this forum is full of problems about how to find last-minute bookings or other mishaps. ESpana as nr 1, then FRance and ITaly as 2-3 and €* to LON are the most cumbersome-the rest is mostly quite easy

I was told that you can take as many train as you can on a single day? Is it possible to take the train from the airport in Milan MPX to the capital then to Bern then Thun and back to Bern.

Then take a day trip to Interlaken and Lauterunnen then back to Bern to then Zurich at the end of the day?

And then I'm going to Germany, Netherlands, Denmark, and Sweden... Back to Italy to NY.

That is correct.

No problem, only that the Malpensa express from the airport is not included.

Yes you can take as many trains as you can on a single day, but some trains needs a reservation. 

Please note, that the Malpensa Express from Milano Malpensa (MPX) to Milano Certosa or Centrale is not included in the Eurail Pass. Normal regional trains from Malpensa should be included, no reservation possible. From Milan to Bern you can take regional trains with change in Domodossola and sometimes in Brig or with the direct Eurocity (EC) with compulsory reservation.

In Switzerland, Germany, the Netherlands and Denmark** and all long distance trains and high-speed* trains are with optional reservation,

 *except Thalys and Eurostar.

** In Summer compulsory reservations on EC/IC trains between Germany and Denmark.

In Sweden and Italy all long distance and high speed trains needs a reservation.

P.S. From Switzerland to the Netherland you could use the night train ÖBB Nightjet (compulsory reservation) and you visit Germany at your way back to Italy. You can find nighttrains also between Zürich-Hamburg/Berlin (ÖBB Nightjet), Berlin/Hamburg-Malmö/Stockholm (Snälltåget and SJ Euronight) and Munich-Milan/Rome (ÖBB Nightjet).