Guys you don't even know how helpful this is. I've been trying to figure this out for months and with your Help I'm almost there in les than 24 Hrs THANK YOU ALL!!!

July 3rd USA to Iceland(3 1/2 days)
July 7th Iceland to Paris(41/2 days)
July 12th Paris to Rome (3 days)
July 14th Evening train to Venice
July 15th to July 23rd Greek isles Cruise round trip from Venice
July 23rd & July 24th Venice
July 25th Venice-Budapest (Visit Relatives)
July 27th Budapest- Berlin (Visit Relatives)
July 30th Berlin- Copenhaguen (Visit Relatives)
August 4th Copenhaguen to USA

By the way. I'm flying out of Paris late on the 12th so will be in Rome 3 full days then taking late night train Rome to Venice

Is this better?

Much better!!!!! Now you've got a trip!

That's looking much better! You might want to check the dates/times though:

<< I'm flying out of Paris late on the 12th so will be in Rome 3 full days then taking late night train Rome to Venice>>

That sounds like you will have the 13th and 14th in Rome - 2 days.
And 5 days in Paris.

Maybe you meant flying out of Paris early the 12th so you will have 4 days in Paris and almost 3 in Rome?

3 days in Rome is not possible, if flights and trains are as listed.

If you are flying out of Paris late on the 12th, and then train to Venice on the 14th, that gives you a max of 2 days in Rome, not three. You will have 2 nights (12th & 13th) and 2 days (13th & 14th).

Even if you fly out of Paris early on the 12th, getting to the airport, wait for security, flight time and getting from airport to hotel in Rome will eat up half a day, so that still gives you only 2 & 1/2 days in Rome. Hope you can get a high speed train to Venice. You need to be realistic about how much time travel takes, and order the days and times correctly. It is better to know how much time you actually have and not overestimate what you have available.

While this itinerary may look better, you took 2 days from London and 1 from Florence, but added 4 for Budapest and Berlin. The idea was to cut, not add. By the end of your trip, you may be exhausted and things turn into a blur. Visiting relatives for just 1 or 2 days can be extremely tiring. Rushing around on a short trip is OK. The longer the trip, the more you need to slow down. Don't underestimate the effect of the heat in Rome and the Greek Islands. IMHO, your trip is doable, but still kind of a race after Paris. At least you have the cruise in between. You will be really busy with sightseeing, but maybe have a sea day to catch your breath.

Better, Rome has a lot to absorb you so you will have to focus on just the things you want to do. Get a map of Rome and start ticking off the must haves, preferably close together or on the metro (using google you can see the little station signs. https://www.rometoolkit.com/transport/rome_metro.htm may help

sassafrass - you and I are wondering the same thing. We need the OP to clear it up!

July 12th Paris to Rome (3 days)
July 14th Evening train to Venice>

sounds like about 2 full days to me but days that are splintered - need at least a day more to leisurely see the main sights of Rome.

The additions all have "visit relatives" associated with them. I assumed that's why she was trying to squeeze more back in.

My favorite itinerary method: list it day by day to be clear how much time you have in each place.

July 3: USA to Iceland
July 4: Iceland
July 5: Iceland
July 6: Iceland
July 7: Iceland to Paris
July 8: Paris
July 9: Paris
July 10: Paris
July 11: Paris
July 12: Paris to Rome
July 13: Rome
July 14: Rome, Evening train to Venice
July 15: dep Venice on cruise
July 16-22: cruise
July 23: arrive Venice
July 24: Venice
July 25: Venice-Budapest (Visit Relatives)
July 26: Budapest
July 27: Budapest to Berlin (Visit Relatives)
July 28: Berlin
July 29: Berlin
July 30: Berlin to Copenhagen (Visit Relatives)
July 31: Copenhagen
August 1: Copenhagen
August 2: Copenhagen
August 3: Copenhagen
August 4: Copenhagen to USA

I would still recommend leaving Paris on July 11, then it looks pretty good.

Since you are going to be in Rome for such a short time, check out Walks of Italy for Rome and maximize your sight seeing. We were there the end of March and wore short sleeves and it got pretty warm in the afternoon. Plan to see something indoors during the afternoons.

Unless the relatives in Copenhagen are extra special I would take time from Copenhagen and add to Budapest and Berlin. I love Budapest enough to keep going back, there is lots to do in Berlin, but I was underwhelmed by Copenhagen.