Well, I'm going to assume you are basically brand new to guitar. Assuming this is true, I'd say you don't have to worry too much. Think of it this way: For your ENTIRE LIFE, your hands have been functioning at a very "primitive" level. All people really do with their hands is open and close them. When you jump on an instrument where you require finger dexterity, it is a COMPLETELY different thing for your hands. It's like a baby learning to walk. Your muscles are doing things that are completely new to them, so they need time to build and adjust. Finger strength and dexterity just come with time and practice. In a month, I'd bet you'll look back at this and wonder why it was ever an issue.

If you get tense and push yourself too hard, you most definitely can injure yourself. Yes, even working tiny muscles in the hand can lead to life long injuries. You should take a break if your hand ever starts to feel sore or cramped. Playing through the pain can lead to permanent injuries in your hand.

What happens when you work STARTING with the first finger? Does it immediately get thrown off when placing the 2nd, or does the 3rd mess it up? Also, still not understanding how your nail is touching any string at all. Hopefully that picture tomorrow will clear things up.