You are not an idiot at all, these are often misconceptions.
There are 2 different outputs on the head unit. The RCA's are considered LINE LEVEL and this is what you want to go to the amp.... This allows the amp to do the work it was intended for. Line level signal is CLEAN (low power) and that's what you want. The other high powered wires are for the head units internal amps and they are usually overrated, inaccurate garbage. These are typically not used at all when an external amp is introduced.
(Unless you are using a line output converter) which essentially strips all the head unit power away and you are left with line level signal anyway.
So to answer your questions:
1. I would like to amplify 4 channels. Is my only option with my current unit to do as you suggested above with the y splitter? If I go that route will I be able to utilize all 4 channels of the amp, or would it only be using 2 channels?
Answer: Yes you will have to split the signal to use all 4 channels to/from the amp. If you desperately want fade AND balance you can use a line output converter on 2 of the head unit powered channels to accomplish this.
2. If I got a new head unit would it come with 4 rca jacks?
Answer: Some do, some don't... it should say in the specifications.
3. Am I correct that adding the amp I have linked (rated for 60w x 4) would boost the per speaker output to 110w per speaker? Basically adding the wattage from the head unit and the amp together? Or... am i an idiot?
Answer: No.. please see above
2 completely different signals.
You also have more options... but I'm not sure you want to go there.. I'm talking about running your speakers in series or parallel. Often used for running multiple speakers on one channel. Then you introduce the challenge of ohms and not dropping them too low in parallel.
Let me know what you buy and I'll send you a wiring diagram on exactly how to hook it up. I'm that bored