I know there’s a lot of discussion of the financial cost to Great Britain in having a monarchy. What I want to know do the other countries that have Elizabeth II as their Queen share in those costs? Or is there only a section in their budget for a Governor-General?
Nope, the only time Commonwealth Realms contribute the costs of the monarchy is when Her Majesty is actually in that realm. So when the Queen goes to Canada it’s the Canadian government that pays for her accomdation, transport, etc, not the British government. I’m pretty sure they also handle the cost of shipping her over from the UK. Also if she’s doing something in her role as Queen of Canada whilst in the UK the Canadian government would pay, but since the only thing she usually does in her role as Queen of another realm while in her home realm is sign letters patent appointing a new governor-general all the Canadians would be paying for is the cost of printing the document and their High Commisioner’s car.
Does the Queen get shipped by FedEx or Royal Mail? 
If it’s overseas it’s via British Airways, since they (alas) retired the Royal Yacht Britannia and didn’t build a replacement.