Building a bike frame up

Why'd you want knobblies on a hardcourt? On a hard, smooth surface, the knobs will lose you grip, not gain it.
Some tread pattern can help if the court has a fair amount of sand/grit on it. Otherwise, look for some fairly slick tires instead. If the other guys are using knobblies I'm guessing it's b/c the easiest available replacement wheels have those on already.

I wouldn't go crazy on tire width either. Width is good for traction on loose surfaces, and (if suitably inflated) will provide a bit of cushioning. Neither are big concerns on a hardcourt. A thinner tire will be lighter, and lighter wheels are easier to get moving again after a stop.

You'll soon find out that buying a bike part-by-part is a lot more expensive than buying one already assembled. And you need some special tools as well. To keep the cost down you'll have to put an effort in at bargain hunting.

It's not as much having to search for parts special to the 1x1, but you need to find part that match the spec that 1x1 is built to. The homepage will tell you all about it, which size seat post, which type/size headset, which type bottom bracket to use. BB's come in different lengths, and length affect the chainline. I can't provide a recommendation there.
I'd be tempted to run a disc brake, maybe even route the cable/hose through the steerer tube. V-brakes would be better than cantis, as they don't stick out that much. Again, inexpensive rim brake wheels are easier to come by than disc brake ones.

You need to watch the wheels a bit too. These days, you can have different axle diameters and different droput width. Check the spec before buying.