OK, this is how I understand it:
On July 16th, every team is given a Mid-level Exception and (unless they used it last year) a $1 Million Exception.
If a team is under the cap, then both of these are applied (even unused) against the Team Salary. Any Disabled Player exceptions, Trade exceptions, etc. are also applied towards the cap
if the team is under the cap.
However, if at any point, the Team Salary (including unused exceptions) drops below the Salary Cap, the unused MLE and $1 Million exceptions drop off the books and the team loses them.
(BTW, the $1 Million exception is a bit of a misnomer, since it will be $1.5 million this year)
Example #1 - On July 16th, Team #1 has $60 million in committed salaries, and the Salary Cap is $42 million. Team #1 gets an MLE for $4.7 million and a $1.5 Million exception. These do not count again the cap, so their team salary is $60 million until they use these (or other) exceptions to sign players.
Example #2 - On July 16th, Team #2 has $40 million in committed salaries, and the Salary Cap is $42 million. Team #2 gets an MLE for $4.7 million and a $1.5 Million exception, which now are both added to the Team Salary. Team #2 now has a Team Salary of $46.2 million, and is free to use either exception.
Team #2 could renounce its $1.5 Million exception (bringing their Team Salary to $44.7 million) without any further consequences. However, if they renounced their MLE, their Team Salary would drop to $41.5 million which would be below the cap, and they would lose their $1.5 Million exception as well.
Example #3 - On July 16th, Team #3 has $20 million in committed salaries, and the Salary Cap is $42 million. Team #3 gets an MLE for $4.7 million and a $1.5 Million exception, which now are both added to the Team Salary. Team #3 has no Disabled Player exceptions or Traded Player exceptions that might also add to their Team Salary. Team #3 now has a Team Salary of $26.2 million. Because this figure is still under the cap, Team #3 loses both of the exceptions before they can use either one, and their Team Salary drops back down to $20 million.
Also, specific to your question, once a team loses their exceptions, they cannot get them back. They are lost for the entire year. So, no, a team that loses its MLE because it is under the cap cannot go over the cap by signing FA's, then get an MLE.
If this makes no sense to you, you can look at what Larry Coon has to say about it - perhaps his wording will click where mine failed:
Larry Coon on salary exceptions