plus the 4 cities you selected shouldn't be too hot anyways, during that time, weather should be on your side.
But, June is the beginning of the "busy season" for sure, you should expect crowds and long lines at the most famous sights, e.g. Colosseum, Louvre,etc.Get Rick Steves's books and there are back doors to avoid some of the lines.
Personally I'm really bias about the tour groups, so I'd recommend that you don't, it takes some planning, but your freedom is priceless:))
Sounds like you have your heart set on 欧洲之星, I took Easyjet from London to Paris, much more inexpensive compared to the train.
Happy Travelling!
