Layover in Rome, What to do?

I have a layover in Rome (coming from Amsterdam to my home country). The layover is from 11:10 AM (landing) to 9:50 PM (takeoff).

First, I never had a layover in a flight, and want to check what is the routine? The luggage will be kept with the airline? After going down from the plane, should I directly proceed to checkout? What are the steps to leave the airport and come back again?

Second, how to go to the Vatican the fastest way from FCO Airport? Note that it will be on Easter Monday 2018. So I've read that by that time Vatican/Rome will be crowded.

Third, I want to check the Vatican and Rome (Trevi fountain, Panteon, etc.). So what's best to do? Go to Rome first then to the Vatican? Or Vatican first?

Last point, at what time should I go back to the airport? Knowing the the plane leaves at 9:50 PM.

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It's normally possible to check luggage through to your final destination but check with the airline and you should have your onward boarding pass as well. The Netherlands and Italy are both in Shengen so there will be no passport control. I don't know Rome but there's a train from the airport which will solve any traffic worries but check the timetable for a public holiday. I would aim to be back at the airport at least 2 hours before the flight. It's probably a bit cautious but you can't take chances.

It depends where your flight originated. If you are arriving from outside the EU, you will have claim your bag, go through passport control and customs, then recheck your bag - normally there's a counter to recheck bags for connecting flights just outside of customs.

If your flight started in the EU, there's no "checkout" and your bag stays with the airline.

In ether case, make sure you have your boarding pass for the next flight with you if you leave the secured area of the airport.

I can't help with Rome transportation, but you want to be back at the airport at least 90 minutes before departure.

GOSH you want to wander around on one of the most crowded periods of the month. DO NOT EXPECT to move fast to the PRIME locations of the Vatican. Extremely crowded during Easter.

Where do you think the Vatican is? It is in the middle of the city of Rome.

You do not depart the Airport the minute your plane arrives.Expect for planing purposes at least 2 hours to get done what you need to in order to leave the Airport train station.
In your case about 13:00 or 1:00 PM

Arrive on the street outside station in Rome about 2:00 PM leave Rome back to Airport BEFORE 6:00 PM Think RUSH hour.
Allow time to get to the train platform in Rome and from the platform to where you need to be at the Airport.

You are expected to be at the check in area for an INTERNATIONAL departure at least 3 hours before departure.

You have about 4 hours to explore your list of sites.

The luggage will be kept with the airline? MAYBE every possible connection you can dream up does not have the same connection procedures.
Your Citizenship?
The Airline into Rome?
The Airline out of Rome?

If you want a better answer give more detail.

Leaving the Airport might require you to have your luggage inspected by Customs and you by Immigration. Before your next flight out you will have the security check.

my home country how nice that narrows it down to about 200 of them. THEY DO NOT ALL HAVE the same procedures for their citizens while in Rome.

Vatican is in Rome. Don't bother with it, if the weather is nice, see the ancient Roman ruins instead.

If your luggage is checked straight through to your destination by the airline then you do not need to worry about it in Rome.

You can certainly leave the airport if you are from a country that does not need a Schengen Visa, but make sure you are back 3 hours before your next flight. You need to allow an hour to get out of the airport and into Rome, plays an hour at least to return, so that does not give you much time in the city.

As others have said Easter is a busy time in Rome. You might be able to get to St. Peter's Square and possibly into St. Peter's Basilica, but the Vatican Museums and Sistine Chapel need to be prebooked and you don't have time to visit anyway. So you might be able to wander around the centre of Rome but first do some research, look at onlone maps and check distances between sites, check public transport options, and work out what you can comfortably see.