River Birch -- if your site is not real dry or clayey, this might be an option to consider
London Planetree - I am surprised this was not on the list of fast growers -- it is basically an imported, less-attractive but less-sprawly version of the native sycamore. Both are pretty fast growers if they have sufficient soil moisture (better on damper rather than drier soil). AFAIK the european planetree tolerates dry clayey soil better than sycamore. They do not usually fall over wholesale but the wood is weak and they lose lots of little branches/twigs, and sometimes calve off large branches. I think they're good looking trees, especially sycamore.
Littleleaf Linden - another reasonable contender, see my previous comments on it
Live Oak - if they're well hardy in your area and will still grow fast enough there, that might be a nice choice as they eventually have a rather spreading shape, good for shade.
American Sweetgum - not the spreadiest tree around, but very attractive. However they will shed those pointy-all-over sweetgum balls every fall, I do not know whether the seeds are poisonous to dogs but a big spread of sweetgum balls is a real pain (literally) to walk over without shoes. Beautiful fall color. Not overly weak wood. Not the fastest grower on the list.
Pin Oak - I don't know about KY, but in the Northeast there is some disease that is seriously killing a whole lot of them. They are moderately fast growers, nice trees, make good shade when *old* (less so til then, though) and tolerant of less than perfect soil conditions... but to me the disease thing would be a concern.
Northern Red Oak - fast growing for an oak (along with pin oak) but still not a FAST grower as such, either of them. Good trees, though. Do you have to worry about dogs eating lotsa acorns? (I dunno)
Thornless Honeylocust - pretty, and moderately fast growing, but not a great shade tree (it is light and dappled shade, not lotsa deep shade). And I can never remember whether it's just black locust, or honey locust too, that has fairly toxic seeds in dog-chew-type pods. You'd want to look up whether this produces seedspods and if so whether they're poisonous.
Tuliptree (yellow poplar) - picky about soil if you want it to grow well, and not really a provider of large areas of shade til it gets big.
Weeping willow - if there is sufficient soil moisture, and no septic system or water lines or house foundation or other paving nearby, and you don't mind what gets squashed by very large limbs (whcih will happen on a regular basis), this is a good option. Very fast growing, good shade, handsome tree. Roots are BADLY invasive over hundreds of feet and can fracture foundations and slabs and paved driveways and totally clog up any water pipes or drains they creep into, so be careful.
Carpathian english Walnut - I would not describe this as fast growing IME, unless maybe you have super perfect conditions <shrug>. Would dogs chew on dropped green walnut fruits?
Quaking Aspen - like honey locust, not really a great shade tree. Fairly good fast grower, though, and attractive. Somewhat weak wood but the branches are mostly thin enough not to damage things seriously, and as it doesn't get as big as poplars, it won't do as much damage when bits fall over or it gets old and has to be removed. Invasive roots (see willow, though it's not as bad)
Frankly if you want shade over a significant area anytime soon, you may need to plant multiple trees, at a reasonably big size. "Fast growing" is relative, and most of these things, if you plant a typical 6'ish specimen, it will be several years before there is ANY pool of shade being cast at ALL, and even once there is, it will only be maybe ten feet across, best case scenario. And will grow only a foot or two per year. So I still think it would be worth SERIOUSLY considering shrubs (which you could afford more of) or vines on an arbor.
Good luck, have fun,
Pat
