What's the weird C thing in notation where the time signature should be? Is it just a shortcut for standard time (4/4)?

It means 2/2. It doesn't make any difference to what's written, it just means that it's to be counted with 2 beats to the bar, if you see what I mean. It's generally used when the tempo is too fast to warrant counting out all four beats (e.g. when there's a conductor).

2/2 and 4/4 are felt differently. 2/2 is felt evenly, while 4/4 is a compound meter where each beat has a different ammount of a accenting. 1 and 3 being the strong beats, 3 being a little less accented than 1, while 2 and 4 are the weak beats.

I'm playing cut time I'm just counting it in 4. It doesn't matter if you say "1 2 3 4" or "1 and 2 and" the result is the same as long as you play the correct feel. I'm well aware of the different feels that's just how I do it haha. I just meant that I find it easier, mentally, to count in 4.

I'm a little confused here now. Maybe I'm thinking about this in a different context but I was always taught that 4/4 is a simple meter where as something like 6/8 is a compound meter. Have I been misled?

This isn't right. Cut time (2/2) is exactly like 4/4, just faster. It would be impossible to tell the difference by simply listening.