Yes some do. I recommend that you follow and contact Mark Siegwart on Youtube or message him on Facebook.
I have not heard of that although anything can and will happen in a nonprofit. Typically, if the organization cannot use regular travel options, they would work with an organization like Angel Flight for emergencies or might get donated travel funds from an airline. More often than not, we are booking the cheapest airline flights available - the ones with no food service and 2 layovers per trip. Nonprofits are always going for the least expensive way to meet a need to maximize how much they can stretch the donor dollar.