Umm...you have left out some really critical information.
What are your actual departure and arrival dates? You said things like your "trip" is June 10-21....you are in Paris 10-13... when you describe your dates like this, it may be unrealistic or not honest with yourself. By that I mean: when does your inbound flight arrive in Paris (and where is that coming from - from London, or Los Angeles?). These details make a huge difference. It's not clear if you will have one day in Paris, or three. If your longhaul flight arrives in Paris on June 10, and you're planning to depart for somewhere else on June 13th, then your entire plan is crazy and unrealistic - because your will be jetlagged, miserable, and you will not have a single, fully usable day in Paris. On the other hand, if your flight arrives on June 9, then you have about one or two usable days in Paris....which many people would say is a tragic mistake (unless you have no interest in Paris, or if you've been there many times before and have seen it all).
Have you been to Europe before? Have you been to Paris, or France, or Venice, or Italy before?
I ask because we see posts every day on this forum from people who have never been to Europe before who ignore the fact that:
You will be exhausted on your arrival day from a long-haul flight, no matter what time your flight gets in, and for most people that day is spent jetlagged and miserable, just wanting to get some sleep. For many people, the next day - their first full day in Europe - is also somewhat compromised, but lets set that aside. You can't count your arrival day (or your departure day) as a usable day - it's not. Because...
It takes time - more time that most people think - to simply pick up and move form one city to another. If you're VERY efficient (experienced) and you travel by train from one city center to another, that may consume just a half-day at best, but in many cases (especially if a flight is involved) it consumes most of a day. Plans like "Day 1: morning in Paris, fly to Venice, sightseeing in Venice" is not realistic - you see/experience neither.
If you are a veteran traveler, if you've been to these places before, then it may be reasonable to plan a trip that gives you just a day or two in Paris....but for most people that's going to be heart-breaking. If you are planing like this, you will have no time to actually be anywhere and see anything, you will always be moving and all you will see/experience is airports, train stations and hotels (briefly).
So...I'd suggest you start your question over. Provide details like when you arrive and when you depart (and have you been there before, plus what kind of things you like to do/see)....otherwise any advice about logistics will be meaningless.
Hope this helps a bit.