I have one at the end of my lane.I only have 1.It gives nuts.

Great!

It's definitely dioecious. But as with most shrubs, it will take a couple of years before it fruits.

Do they really grow 15' or even 18' tall? I want them to grow tall.

They do grow that tall. They will also grow tree like if you remove all of the suckers. They provide a very dense shade.

C.americana is monoecious - both male and female flowers on the same plant, but for the most part, they're not considered to be self-fertile - you really need two different plants to get reliable nut crops. Bulldinkie may have a clump that's actually more than one plant, or there's another one nearby, if she's getting well-filled nuts.(Be aware that they're tiny - the nutmeat in a big one may only be about the size of a (small)green pea, enclosed inside a fairly thick nutshell, and you'll have to beat the local squirrels to them.)
They grow wild on the farm here - I've never seen one more than 8-10 ft tall, max; most are smaller. Multistemmed shrubs - yes, I suppose you could force it to be a single trunk, but you'd be forever cutting back new suckers at the base.
If you want a single-trunk small-medium tree, look at C.colurna.

They are obligate outbreeders, meaning they require another plant's pollen to fertilize their flowers. Bulldinkie's shrub was likely pollinated by a nearby plant, as the pollen is wind-dispersed and can travel great distances in the right conditions.