I'll address your questions in reverse order:
Many teachers will hammer it into beginning drummers that counting out loud is an absolute necessity, and for some people it is. If you have no musical background, counting out loud is a way to embed the habit of rooting yourself to a beat pulse. That said, if you have no problems maintaining the count in your head, more power to you. Now, is it necessary to always count, no matter what? This is often opinion masquerading as fact, but for the most part, yes it is. That doesn't mean that every drummer you see is always counting in his or her head. Over time, through counting and practicing, you develop feel. You can feel the beat and the count even if you aren't actually counting it. That was something that came rather easily to me when I was young. You learn to feel where the downbeats fall and how the syncopation and offbeats meander around the pulse. Unless you're freeforming, being able to stay rooted to a pulse is essential.
As for developing limb independence, as you said it comes with practice. There really aren't exercises, per se, that develop it, it simply kind of flows from developing your technique. You can stumble for months on the most basic rhythms until you buid up the muscle control to fluidly maintain them. Then, you throw an odd beat or two in there, fuck yourself up all over again so you can build up more control. Gradually, as you try different things, you'll find that you won't have so much trouble with newer rhythms and tougher compositions.
Practice rudiments to develop stick control. Work on basic rhythms and small riffs to build up your fluidity around the kit. Coordination and independence will come with time.