How to create a photogate that can span tens of feet with multiple sensors to capture items of various sizes passing through it?
If you need to detect a six inch object passing anywhere in between the columns you'll likely need to use lasers. Otherwise an object passing near the emitter side might go unnoticed because of leakage from the other emitters. You can make that easier on yourself (in terms of alignment and expense) by using visible light lasers, just some red laser modules for instance, like those found in inexpensive laser pointers. You should probably also place them every three inches vertically. Of course if you're detecting dogs doing agile things, like the use of the poles in the photo, six inches is probably padded by quite enough already.

For the detection circuitry it might be better to know which detector is being activated. This way if two objects pass through at the same time you'll have the ability to detect that. If you know you'll have 3D things that when viewed from a 2D surface will trigger at multiple points as they move through, you can just OR your inputs together. I would use shift registers for this. Gather eight values at once into a serial stream to an MCU.

You have the right idea for ignoring sunlight/erroneous signals. Modulate the light at a known frequency. You can probably leverage IR remote control circuitry for this, that it typically modulated at 44kHz. You may be able to get an IR receiver that has a low Q in terms of light detection, meaning it would activate from a red laser. If you decide that this isn't too difficult then you could try for emitter/receiver frequency pairing system you suggested and avoid lasers all together. 

It's much cooler, still after all these years of common use, to say you're building something with lasers though.