When do Japanese high schools have midterms?
Your quoted info is right.
Most high school have first mid-term exam around May 20th.

The problem is magazine system.
As you think, most magazine's release date of issue 7 (July issue) is some day of June,
because no one buy July issue in August.

But some magazine have more wider shift from titled date.
JMPA (Japan Magazine Publisher Associates, which most Japanese magazines are members of) has the less than 40 day guideline for monthly magazine.
Publisher (and book store) can sell magazine 40 days before the titled release date or month.
(Weekly magazine have 15 days guideline.)

If the titled date is July 1st, publisher can sell the magazine after mid May.
Of course, this is most extreme case, but it's possible that people can read issue 7 in May.

To evidence, you can find example from Amazon.co.jp magazine corner.
For example, this magazine is July issue (07月号) but Amazon starts sending that magazine mid May (2014/5/17).
(Yes Amazon starts earlier than 40 days rule, and some book stores are angry about this, but Amazon says customer will receive the magazine after the date).

I think fashion magazine has earlier selling start date.
Using the above example, the magazine features topic on how to select swimwear, which most people prepares before summer but uses it in summer. The date rule fits the use case.

Weekly Shounen Jump is officially release on Monday, but some store selling it from Saturday.