Is it? I did a google search and all the references to C-130s that I found had a capital C, but I didn't find any written rules about it. I would feel better if I had some confirmation.
If all the references have a capital C, then you won't go wrong if you use a capital C. I've never seen it written any other way, so I say use the capital. I don't know of any formal rule about it, though. Frankly, it would never have occurred to me to ask.
Yes, the C is capitalized. All aircraft with a C prefix are referred to as transport craft today, though the C is originally for cargo.
I've always seen it capitalized. All helicopter letters are capitalized too. AH-64, UH-60, etc. I've never heard a specific rule on the matter but I've never seen it any other way.