My son is graduating from high school. Is it proper to send invitations to people who also have a child graduating at the same time?

Yes it is okay but what often happens - if they are all in the same graduating class, is that the Open House invitations are sent in lieu of the graduation announcement. With so many Open House gatherings to go to, parents are opting for the day before graduation or even the day after. Nice to keep them all straight!

Do you mean people whose children are graduating from the same school? If so then it's obviously not necessary since they are already going to be there. If it's people whose children are graduating from different schools, then try to be mindful of any conflicting times and send no invitations to the people whose childrens' graduations conflict with your own.

Yes it is perfectly fine as you most likely will be receiving one from their parents/parent also. In fact they would probley be upset and hurt if they didn't get one .

I'd make the invitations for the party only and not mention the ceremony on them. I think the other parents of graduates are kind of like "co hosts" of the event in a way, so should not be invited to the ceremony additionally by you.

Invite the guests who are included in the ceremony verbally - my experience is that usually only immediate family and very close people like a best friend or girlfriend/boyfriend are invited to the ceremony. It's not like a wedding or Christening - it's perfectly appropriate to invite people to just the party. Very often, graduates are limited to only a few tickets for the ceremony, so it's not even possible to invite all the guests to the ceremony.

I would invite them not every kid has there party right after graduation I remember when I graduated there were parties like 2 weeks before we even had graduated.