Mmmm…..   So are you looking to set up something permanent or collapsible that you set up each time they have outside time?  I thought something like this if you could get 2nd hand or adapt a gazebo to suit.  Though don’t think it would withstand a cyclone!

Obviously it doesn’t have to be as tall as this unless you want to hang out in there too, away from the mozzies.

Another ultra cheap way to do a canopy is iron droppers, poly pipe and netting.  You put droppers into ground then put one end of a length of poly pipe over end of a dropper then forming an arch, put other end onto dropper opposite. Do a row then throw fine netting over the framework. Again, not really stable but quick and easy.  

Edit: Found some pics that sort of show this. Used for netting fruit tress etc.  Looking at them now though, they do look a bit flimsy!

I think large form grid on ground could work.  Feet would be protected if grass is long enough to come thru  it.   Be careful though because of bugs that live in the grass and if feral cats pass thru the yard, that might bring some nasties too.  Since you’ve got some time before they arrive, what about sowing some lawn seed in a specific area and keeping that out of bounds for any animals passing thru?  You could even put some weed mat down, then soil, then seed and if there’s any digging, it would only be down to the weed mat.

Another thing is be on the look out for second hand rabbit run or bird aviary. With a tall aviary, you could put it on it’s side to make more room horizontally.

What about a covered dog run like this? You could buy one-or even fashion one in the same idea

*gulp* Birds of prey and cyclones?? 0_o

Might I suggest a quonset hut instead?

You can actually find smallish ones, like this:

Although the sides and ends would need some re-inforcement.

Well, why don’t you go for a triangle with a longated rectangular base. The walls would be pointing up, into a triangle. your could provide shade by placing it next to a tree.