Can "Second" be used without "First" in an earlier paragraph?

Obviously it can. You gave an example of it.

I suspect you are asking...should it be done.

My answer would be that stylistically, it would be best to use "First" in the first sentence...and then use "second" (or secondly) in the next one.

It's sorta like starting a conversation with, "And in the second place..."...which I did one time. I had been thinking arguments over in my mind before opening my mouth...and sounded like an idiot. We both laughed at my mistake.

It sounds terrible. It also doesn't sound like it is in the same vein as the first sentence. If you are numbering something (like thoughts), you would think that each sentence is a thought. Here it sounds like the second sentence is a clarification of the first, not something of equivalent value.

Yup.

The paragraph just doesn't have a decent tone to it as written.

I would have replaced "Second" with "Also". I agree with you two about the numbering. I mean, how is it any different if you had no First or Second, and started your third point with "Thirdly"? Weird. It would make me want to mentally insert a Firstly and Secondly and that would just annoy the crap out of me. Don't make me edit your stuff!!! Plus, consistency is the key.

I have a feeling that there are previous sentences that have been left out.