I never played Spelunky classic. I bought the game a week ago and am enjoying, but something doesn't seem right. When I pick up a gun or a camera, I have to set it down to use my whip. However, I've read online that using 'C' should cycle through your items. 'c' does nothing for me. Can I only carry one thing at a time?

It's been a while for me, especially with the freeware release, but in the remastered version you definitely can't use your whip while holding an item.
I think the cycling of items was/is only about switching between bombs and ropes, which I think isn't possible in the Remaster.

What you read only applies to the original; the remake has no way to cycle through ropes/bombs/weapon (and, therefore, no way to drop unlit bombs to trigger traps, then pick them back up, which also was a thing in the classic version).
And yes, you can only carry one held item (rock, damsel, camera, etc.) at a time.

Yeah. The onscreen instructions. Check the lower right for a prompt.

Late reply, but the default keyboard commands for character selection are the left and right arrow keys, with the Z key confirming your selection (and the S key being the only way to deselect/go back via the keyboard, despite no such prompt being shown). EDIT: Also, if you change your movement controls in the options, the character selection controls will change, too.
As much as I love this game, the controls in the menus are pretty inconsistent.