Is summer event going to start immediately after anniversary?

you can look up the "fgo jp event list" and compare the dates 2 years back. other than that we don't know

At the moment, all signs point towards it starting on the 18th, but nothing has been outright confirmed as of the current time.They may add a week in between, but I'm inclined to say the 18th, at the moment.

There was a 2 week delay between Anniversary and Summer in JP. So unless NA decides to speed it up, it should arrive around late July or early August (2 weeks from the end of NA anniversary is 1 July).