There is an excellent cafeteria near the gift shop but I don't remember that there is an Italian dish. Nearby, though not close, is Monroe St which has a number of restaurants. Not sure any of them are Italian, though.

We stayed at the Capitol Skyline...close to rail line too.

The Shrine is not near many hotels. DC is outrageously expensive. I’d stay at BWI and take the train then Uber or Lyft from Union Station.

We parked on the street and there was no limit. The White House is quite a distance from the Basilica and it would be difficult to time it right to meet your entry time at the WH.

Of course. Many groups come to the Basilica to visit and tours are given as well. There is a cafe and gift shop too. It is a big place.

Moving around in the Washington area between 4 to 6 pm is always a challenge. Public transport let's someone else do the driving.

Yes there is an elevator

It would be possible to visit both in the same day. You could arrange a tour for one in the morning and the other after lunch. I do not know if either place has picnic facilities. But you can get a reasonable cost cafeteria lunch. I know this is available at the Basilica but am not sure of eating at the Franciscan Monastery. Both places have excellent parking for buses.

We went for Easter Sunday Mass. what we did was get there early and tour around outside and in the crypt/chapels/artworks underground while one of the earlier masses were on. Then we went up just as the mass finished and got a seat right at the front. After mass we explored the rest of the basilica. By doing that you are being the tourist and the good Catholic at the same time and not wasting your time.

Yes it is beautiful. The noon mass has beautiful music.