First of all, unlike Spider-Man, Captain America does kill. He's a soldier, after all. He'd prefer not to, but if he's in a dangerous situation with innocents at risk (like a war) then he will kill if he has to.

Secondly, if he were to have a gun, I think he'd only have a small pistol that doesn't encumber his movement. Since while he's no Spider-Man, he's still very acrobatic in his fighting style.

I second that shield combat has to be a top priority. The shield has to be as versatile and natural to use as Spider-Man's webs are in Spider-Man for the PS4.

Ultimate Cap uses all weapons (good ole edgy universe). 616 Cap generally doesn't use fire arms but might use what he finds on site in an emergency. MCU is a mix, having brought and used small fire arms, but mostly restricting himself to the shield and his fists.