I know that 1 USD=12 Peso (or something around that).
So does that mean that 12 peso are 12 pesos? Because the websites I was looking at only used the word peso in singular form. In other words if you exchange 1 USD for 12 peso, you can spend those pesos independently, right? Like you can pull five pesos from that 12, and it leaves you with 7?
I… I don’t understand your question. 
1 USD is approximately 12 mexican pesos using the current exchange rate.
In English (and I believe in Spanish) the plural of “peso” is “pesos”, so the website you used had a grammatical error.
I am not sure what the rest of your question means. I suppose if you are asking if “12 peso” is a different currency from “Mexican pesos”, then the answer is almost certainly, no, it is not.
In both Spanish and English, the plural of “peso” is “pesos.”
If you show up at the airport and the dollar purchase price is MX$12.00, you’ll be given 12 pesos. That may be a 10 peso coin plus a two peso coin, or any mix and match thereof. If you use the 10 peso coin to buy a bottle of water that costs 5.50 pesos, you’ll be given 4.50 pesos as change.
Yes