Whether to schedule the trip around Le Mans at the beginning or end of the itinerary hasn't been fixed yet, but we are indeed targeting this trip around this event one way or another, so we are not looking to schedule it for other months in the calendar.
For me, either arrangement is fine, but what has been lacking is feedback. I'm trying to organize this around YOU guys, not me.
June is still not bad with regard to vacation crowds in Germany, since school summer holidays generally run from mid-July through August. On the other hand, most people in North America also take their summer vacations in July starting with the 1st (Canada) or 4th (USA) of July holidays.
The only thing to keep in mind is that the Le Mans race itself starts on Saturday afternoon and ends 24 hours later on Sunday; to make the most of the weekend, you should also plan to arrive on Friday at the latest and don't plan to rush to catch your flights home right after the race on Sunday (from Frankfurt, no less, 760 km / 470 miles away) so that you can got back to work on Monday.
From the standpoint of the destinations we would want to visit, it would be seem to be best ot schedule the trip with Le Mans at the very beginning or end because it is the geographic outlier. A third scenario of having Le Mans in the middle of the itinerary is also possible. This would seem to be near ideal, since we would start on June 16th, end (officially) on the 30th, and people can then choose to stay on longer through the 1st/4th of July holidays or go home to recover.
Again the lack of feedback hampers figuring things out further.
Sharing one rental car among multiple people is certainly OK if such groups who are willing to do this can be formed.