connection time needed

The first thing I'm going to suggest is buy all of it as one ticket. This way, if you do miss a flight, the airline is obligated to get you on the next flight. If you have checked bags, they will automatically be transferred to your next flight. If you buy separate tickets, this changes everything.

At DFW, all you will have to do is change gates.

At Rome, you will have to go through immigration into Schengen.

In Dublin, you will go through US immigration and customs prior to getting on the plane.

When you arrive at PHL, it will be like a domestic flight and you just have to change gates.

As for how much time, see what the airlines offer as connections. They are--usually--doable. But not always.

I recommend at least 2.5 hours for the international legs of the trip (DFW to Rome & Dublin to Philly) and 1.5-2 hours for the domestic legs. That’s my comfort level; I’d rather hang out in the airport for a while rather than rushing to make a connection. The trip sounds fun! I met some folks who took that trip and they loved it.

I prefer going as directly to Europe as possible first, then switching to an inter-Europe flight. Seattle has direct Amsterdam flights that fit that bill. Allow at least 2.5 hours to transfer at Schiphol, a little more would be better to ensure you get where you need to go. If you choose that flight routing through Dublin, give yourself more time as you go through US border control process there (so you don't have to when you get to the US). You might get better feedback if you list out your options with connection times so people can weigh in on preferences.

Note, there are new paperwork requirements if you fly through a UK airport, even if you don't stay there. It's not cumbersome but it requires planning, see post with info here https://community.ricksteves.com/travel-forum/england/uk-eta-application-experience

That’s a lot of flights. You should be able to do that with just 1 stop each way. Not sure what your travel dates are but there are new flights out of SeaTac on SAS - through Copenhagen - and they are pricing out very inexpensively for many dates this summer. I just dummy-priced $750 round trip in August - non-stop SEA to Copenhagen.

I like Kathy H's answer

I recommend at least 2.5 hours for the international legs of the trip
(DFW to Rome & Dublin to Philly) and 1.5-2 hours for the domestic
legs.

We try to leave 3 hours for connections. If your flight is delayed departing, it takes a while to taxiing to the gate, then it takes an hour to get to you next gate due to getting off the plane, walking, and passing through security, you’ve used up a couple hours. Then, you need to be at the new gate 30 minutes before departure since that’s when gates close.

From my recent international experience since October, two flights departed 30 minutes late, three departed 1-1.5 hours late, and one departed 2.5 hours late. While my two domestic flights departed the gate on time, one of them was held on the tarmac for a hour by the tower due to traffic congestion.

Simple answer: if ITA Matrix flags a "tight connection," pick a different flight.

Well, I am a Delta flyer leaving from the Spokane airport so have to transit thru Seattle or Salt Lake City.

I'm also assuming this is the 14 day Best of Europe tour?

I'd do the Seattle -> Amsterdam (Delta), Amsterdam -> Rome (KLM codeshare) outbound. There will be a bunch of KLM CityHopper flights from Amsterdam to Rome, probably every couple of hours in case something goes sideways. Returning home, Paris CDG to Seattle on Delta.

I far prefer to change planes outbound in a European Hub. If you change in the US sometimes there is only one flight a day to the European destination so if it is delayed or cancelled it throws everything off. With a transit thru a European hub there are usually flights every few hours to the big European cities.

I hope you have enough time/budget to go at least the day before your tour starts, if not 2 days before! IF you can stay in Paris longer, definitely do that!

Depends on the airport.

We fly Delta a lot and at Atlanta one hour and a half is good.
In the rare occasions that we would change flights in Wash-Dulles or JFK, we would go with 2 hours. Same for Chi-O'Hare and LAX.

For Paris-CDG, 2 hours minimum, even 2 1/2 would be better, same with Heathrow if you have to change terminals.

DFW have only been to on a direct flight, no changeover.

My home airport is Seattle and I generally fly Delta (KLM/AF codeshares) since I like their flights and pricing. Their connections are generally 90 minutes thru Paris (CDG) or Amsterdam (AMS). Ideally, I would like longer connections, but what is ideal and what is a good price are rarely offered together. In my opinion, 90 minutes is sufficient for AMS. Can also be done for CDG, but for that airport, I would "walk" the route ahead of time and walk very quickly. For tight connections, try to sit closer to the front of the plane, and arrive earlier in the day, so if you miss the flight, there are later options to get to your destination at a later time (check to see how many options are available for later).

thanks everyone. Sadly I have credit with American Airlines, so that's who I was looking at...but I'll give Delta a try.

You could try booking partner flights through the AA website (usually you can use credits as you are booking through them.) There are non-stop flights on BA (SEA to LHR), AerLingus (SEA-DUB) or Air Tahiti Nui (SEA to Paris--not sure which airport but think it's CDG), which are all part of the OneWorld Alliance with American Airlines. SEA is our home airport and we typically fly Delta/Air France/KLM. You might also check flights from YVR (Vancouver Int'l) as their prices can be significantly cheaper. Best of luck!