My first recommendation would be to make sure using a carousel for your content type is the right idea. I say this because user interaction with carousels is typically extremely low. Auto-rotating carousels have slightly higher interaction numbers, but not much. So if you're placing important content within the carousel...be sure it's on the first "slide".
In response to your actual question regarding navigation I would recommend you absolutely keep the navigation arrows. There needs to be blatantly obvious indication that this is a carousel to the user. Not sure if you have pagination indicators as is typical, but those generally are not indication enough.
As for your concerns for space if keeping the arrows just adjust the positioning of the arrows in your media query. Maybe move them so they're right below the carousel content. Perhaps position them absolutely so they're z-indexed on top of the carousel with a lower opacity. Both of these would not effect the amount of horizontal space required, and one would require a bit more vertical space depending on the dimensions of your arrows.
For touch devices enable the ability to swipe through the carousel. I'd also recommend enabling the ability to "peek" which is when a user drags just a bit and can see that there is another slide to the left/right.
Here are a few great links about carousels, their stats, patterns, best practices, etc.