For training with a grace month, is it legal to fly in the month after training expires before going to school?
Yes, the FAA issued a Letter of Interpretation from their Chief Counsel in May 2015 which covers Section 61.58 proficiency checks and specifically addresses this issue. 

It says that a pilot may continue to act as PIC during the grace month under Parts 91, 121, and 135.  It says (in part, and emphasis added by me):




  Section 61.58(i) states that, if a pilot completes the proficiency
  check in the calendar month before or the calendar month after the
  month it is due, "the pilot is considered to have taken it in the
  month in which it was due for the purpose of computing when the next
  pilot-incommand proficiency check is due." 
  
  You have asked whether the language in paragraph (i) creates a "grace
  month" during which a pilot whose currency has lapsed under paragraph
  (a) may continue to act as pilot in command. The FAA has previously
  stated in a legal interpretation that § 61.58 includes a "grace month"
  for purposes of complying with the 12- and 24-month proficiency check
  requirements. We did not explicitly state, however, that a pilot may
  continue to act as PIC during this grace month. Legal Interpretation
  to Sean Conlin (Feb. 24, 2000). We have indicated, however, that
  similar provisions in parts 121 and 135 create a "grace month" during
  which a pilot may to continue to act as PIC during the month after the
  pilot's currency has lapsed? Legal Interpretation to Gregory S. Winton
  (2006), Legal Interpretation to J. Dennis Vanatta (June 25, 1996). 
  
  The FAA intended the grace month to limit disruptions in operations
  conducted under parts 121 and 135 by accommodating pilots who are
  unable to complete recurrent training requirements in the month they
  are due. At the same time, the FAA did not want operators to use the
  grace month as a means of regularly extending a 12-month proficiency
  check to a 13-month proficiency check. For that reason, a proficiency
  check under part 121 or part 135 that is completed in the grace month
  is considered to be completed in the month it is due. The FAA believes
  that the same approach is appropriate for§ 61.58 proficiency checks.
  As such, a pilot may continue to act as pilot in command during the
  month after a§ 61.58 proficiency check is due. However, when a pilot
  completes a §61.58 proficiency check during this grace month, it is
  considered to have been completed during the month it was due for the
  purpose of calculating the next proficiency check.