Use Allegro 5! Its sooo easy to make games in if you know basic C++. Here are some excellent youtube tutorial playlists:

https://www.youtube.com/playlist?list=PL9333715188CD7669

https://www.youtube.com/playlist?list=PL6B459AAE1642C8B4

As for your questions

1) Ive only heard of SDL being used when Valve was porting a game over to Linux to handle input and windowing

2) Yes an engine takes care a lot for you and limits you as well. Use an appropriate engine that fits with your game. On contrary to your first question,

engines such as unreal and unity are very popular when it comes to professional game development

3) Sure you can opengl without sdl/allegro/sfml but why would you want to? Those three libraries make opengl so much easier to use.

SDL 2.x is excellent. In retrospect I wish it were available when we'd started work on our engine, because it would've probably saved me a bucketload of work. The 1.x branch available at the time was not suitable for modern development. They've made great strides with the new version though. My experience so far is that I'd probably be opkay with shipping it in a pro level title, though saying it and doing it are two separate things.

You can make a game without any libraries at all, if you so desire. (Apart from the underlying operating system and graphics API stuff, which is necessary.) There are reasons to do this. They probably do not apply to you.

As for your first question, if you are sticking to C or C++, I'd also recommend SDL 2. It is extensively used in the games industry, as the main development technology or sometimes to port games to new platforms.

About the engine controlling your game, that is not really accurate. You can do whatever you want with an engine like Unity, it does not restrict your ability to implement the gameplay. Of course, if you need some super fancy features as an exceptionally accurate physics or lighting, you'd be in for a hard time; it's hard to see a case where Unity wouldn't suffice though.

Now, about the advantages of using it, it is not an easy analysis to do. You have to consider the price of the engine for your desired platforms. Let's say your game has only simple physics, maybe some neat lighting system and doesn't need all features from Unity, chances are you could actually develop a smaller engine with a much better cost (time-cost) than you'd have to pay for it (monetarily). That is really case-specific.

1) Ive only heard of SDL being used when Valve was porting a game over to Linux to handle input and windowing

I can name many commercial games made with SDL over the top of my head: Starbound, Prison Architect, Amnesia, Faster Than Light, Overgrowth, Second Life... Not to mention that PyGame, Löve2D, Cryengine and many other engines are actually built on top of SDL.

I can name many commercial games made with SDL over the top of my head: Starbound, Prison Architect, Amnesia, Faster Than Light, Overgrowth, Second Life... Not to mention that PyGame, Löve2D, Cryengine and many other engines are actually built on top of SDL.