Knowing that we are a little late with our planning, we are about to book one week's vacation to Arizona in early April this year. We travel from NYC and are 2 adults and 3 children aged 7, 13 and 16 years. We will rent a car from Arizona.
We have 7 days available and would like to visit the Grand Canyon, Mesa verde and Antelope Canyon as well as Lake Powell if possible. The petrified forest is also on our wish list but not a must see for us.

Is there anyone here who has made a similar trip and can help us assess whether it is at all realistic to reach the above destinations in 7 days and whether it might make better sense to fly to either Las Vegas or Salt lake City as a starting point for this journey.

We we be very grateful for input

Just wanted to warn you that most services at Mesa Verde do not open until April 14th. In addition, both Balcony House and Cliff Palace are closed until road work is completed around July 1st. Check the website. It's a wonderful place but you may have to scale down your expectations just now. https://www.nps.gov/meve/index.htm.

The most efficient way is to fly into Phoenix as it is only a little over three hours from the Grand Canyon. Spend a night or two and take a couple of hikes and then go to Antelope Canyon (access only by guided tours approved by the Navaho Nation only). Then travel to Monument Valley Tribal Park and stay at the Vista Hotel. Finally, drive to Durango and the next day go to Mesa Verde. On the return trip to Phoenix you can stop off at Petrified Forest. Lake Powell would be out of your way and the water levels are at a record low.

I'd definitely take Mesa Verde off the table for the reasons Fra_Diavolo mentions and because it would add a good 6 to 8 extra hours drive time.

You could do PHX > the GC > Lake Powell/Antelope Canyon > Monument Valley and back to PHX - it would be a busy week but doable. You are very late planning. But they do get daily cancellations at the Grand Canyon so you have a semi-good chance of getting in.

I agree. If lodging at the south rim are unavailable or too expensive, a motel in Flagstaff works OK for taking day trips to GC, Sedona, Winslow and even Monument Valley. In November 2019 my wife and I flew to Phoenix-Mesa on Allegiant and rented a car in Mesa (non airport) for our trip with the south rim as our main focus.
If you decide to fly to Las Vegas, the rental cars (and pickup trucks) are a bit cheaper than in the Phoenix area. With Las Vegas, you also have the chance to see Hoover Dam and maybe even Valley of Fire SP.
South of Sedona in the little town of Cornville AZ we had a great anniversary dinner at the Manzanita restaurant. They specialize in authentic German food and some wild game dinners