You can lay your wire mesh or whatever on the ground rather than digging. I've read 3' is safe though I'm sure some only go two. Of course it must be well secured to the coop and on the outside edge -- I gather there is some sort of lawn staples you can do this with. In time the grass will grow through it and secure it. I like this better because it's so much easier to check or replace if needed, besides being a whole lot easier to install. Of course there are also things like paving stones but that can get expensive and probably isn't as secure.
Screws holding the wire will be much more secure if they first go through a wide washer with a screw sized hole in the middle -- we used to call them fender washers.
You really don't have to worry much about draft in that climate, and I would only insulate for coolness, not warmth. Chickens have a whole lot more trouble with heat than cold and will need breeze almost year round. I think if it's hot enough, it's best to think breeze rather than draft. A 3 sided coop is plenty in hot climates. They will appreciate if there is no draft directly on the roost in January, but other than that it is better to be too airish than not airish enough. I would probbly put up some tarpaper and leave it at that, and not in the whole coop, just where the roost gets blown on. Not sure whether you would need OSB or something over the tarpaper to keep them from tearing it up -- probably you would.
I'm not sure that pigeons in there 10 years ago would pose any danger at all. Personally I'd spray a dilute bleach or ammonia solution around and call it done, unless I could find info to the contrary. Ammonia has the advantage of killing cocci which bleach will not, and yours is a wet climate, so I would certainly spray some dilute ammonia on the ground. There is a good disinfecting product you can buy called Oxine, but it costs.