I don't know what tapping is even after reading a definition and I think splitting is what the push/pull does,
One component of a guitar pickup is a coil of wire. The guitar string vibrating through the pickups magnetic field induces a current in the wire coil. The more turns on the coil, the more current is induced and the pickup can push the amplifier harder. More windings typically add more midrange and reduce top end.
If someone wanted to reduce the output of a coil, they could "tap" it part way down. So, if a coil had 6500 turns would around it you could tap it at the 5500 turn mark and get a drop in output and a shift in tone.
A wire coil is also known as an antenna, and a single coil can and will pickup unwanted background noise. A humbucker has two coils as is designed so that background noise picked up by the one coil is cancelled out by the same noise being picked up by the other coil (ie, it bucks hum). It also changes the tone because two coils are sensing the string from a slightly different point on the string, and then the results of both are added to each other.
A coil "split" is when you split a humbucker and only use one of the coils.
PRS uses a tap. Their humbuckers are setup so that one coil stays on full, and the second coil is tapped for reduced output. This allows you to get a quasi single coil sound from the one coil, while retaining the majority of the humbucking from having the second coil still somewhat active.
I don't know if the double or single cut changes the sound or is just a style preference either or if having extra volume/tone controls would be that useful.
Generally, a single cut is considered to produce more low end than a double cut. That said, my McCarty is thicker and smoother sounding than my SC, and my SC is much brighter and more cutting than my McCarty.... So wood plays a role here too.
Wide/thin, fat etc...? Also I think I am leaning toward a 22 fret model now too. I am a total novice in the electric world but am learning slowly.
Thinner necks are generally preferred by guys who play fast and shred up and down the neck or play with the thumb on the back of the neck. Thicker necks are generally preferred by guys who play slower or play with their thumb hooked over the neck. That said, it's a pretty personal decision and there's probably a lot of people who disagree with what I just wrote.
There's a bunch of differences that come with 22 vs 24 fret, but the one most focus on is the change to the neck pickup tone. It's a little thinner on a 24 fret guitar. I prefer 22 fret guitars for the fatter, warmer neck tones, particularily when playing clean. However, the thinner sound of a 24 fret neck pickup seems to work better when using high gain and soloing. It stays clearer and more defined were as the 22 fret neck pickup will typically get a bit muddy.
If you're undecided/unsure, most people use 22 fret necks. If you want fatter clean tones and/or play bluesy rock you'll probably be happier with 22 frets. If you're thinking you might want to take up lead and start shredding solos at 1 million notes per second then you'll probably be happier I with a wide/thin 24 fret neck.
Why does the S2 CU get your recommendation over a SE CU?
Probably because the S2 is a "better" guitar.
SE guitars are good value. I own one SE and two Cores. I in no way feel that the SE is a cheap guitar. It sounds great and plays great. But compare it to a Core and its obvious that the Core is several steps up in terms of quality. It's not just "bling factor" either, there are many functional and material upgrades to the guitars as well.
The S2s are in between. A step up from SE, and a step down from Core. Since you appear to be looking for a workhorse guitar I'd suggest either an SE or S2, depending on your budget.
Here is a short list of some of the guitars I am looking at still. No particular order.
SE Custom 24 (Floyd)
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Unless you're planning on doing dive bombs and/or heavily abusing the trem you probably don't want something with a Floyd. See: Van Halen, Kirk Hammett, Those dudes from Dragonforce.