Morning walk to the Vatican

1. Yes, it will be quiet and lovely. I've done early morning walks and I agree this is a great time to enjoy the streets/views. And the light at sunrise in Rome is beautiful!

2. I don't really know - I've come from the south (Trastevere and Navona areas). It's probably out of your way, but crossing the Ponte San Angelo is the prettiest approach to the Vatican, in my opinion.

3. Yes. Some places will be closed, but others will open at 6. I am not sure about 5:30, I'd bet you could find something, but by 6AM you will have a few choices. If this is on a weekend you might have more trouble than on a weekday.

It will be extremely quiet then...next to no traffic, very few pedestrians.
I also stay in Trastevere and it's a quick walk from there.... it shouldn't take much for you to figure out from your location if you have a map.
You'd best plan to have a cup of coffee and something to eat before you start out. Next to nothing will be open that early. There are a few bake shops in the neighborhoods adjoining the Vatican but you'll likely have a hard time locating them.

From Piazza del Popolo, head to the east side of the Piazza climb to the Pinico Terrace. Great views over Rome. Back down to Via del Corso, past the Spanish Steps (there is a McDonalds) step in to see how Rome does it.) to the Trevi Fountain then wander the streets heading west past the Pantheon to Piazza Novona. Head out the north side of the square, walk along the river to Ponte Sant'Angelo

Use a good paper map, or use google maps

Edited to add it's about a 2 mile walk.

Great idea! Any Italian City — and many in other countries — is delightful as it wakes up!

You have two basic route choices, IMO

1. Cross the river at Ponte Regina Margherita, which takes you down Cola Di Rienzo, a major shopping street. It will be dead at that hour, of course.
2. Walk down Via di Ripetta to Ponte Cavour and cross there. Via di Ripetta is quite nice, a mix of shops, and the bridge is right by the Ara Pacis. This route will take you past the impressive Supreme Court building and Castel Sant’Angelo.

Use Google Maps to refine your path beyond getting across the river. The area is littered with bars (Italian coffee shops) and some will open as early as 530 for early workers, more at 600, and many more at 700. More cafes will be open on sidestreets so wander a bit.

Terrific idea. have done it for years, however, I stay at a former convent close to the Vatican so I can get to St Peters' when it opens at 7 for daily Mass, then find go to a local Taverna for sweets and coffee.

What time do you recommend being in line if we want to attend daily mass?

There is hardly any line at 7. What you do is enter St. Peter's, bear left and look for the entrance of the Sacristy, which is about at the middle of the left wall. There will be several Priests' exiting, and you will usually hear an English speaker as he comes out, or his family might be waiting for him; just follow him to his assigned Altar and attend Mass.
A bonus of going that early is there are no barriers set-up yet in the church so after Mass, you will be free to wander all over, even into areas that are normally blocked off from visitors. Enjoy!