So if you roll a 6 first, what's the chances of rolling a $6$ again with the same dice on the very next roll?
I understand $100\% / 6$ numbers $= ~15\%$ chance, but does the roll count factor in here? If so, does that double the chance of rolling a $6$ again for $~30\%$ chance or half it for like $8\%$ chance?
Dice have no "memory", so the chances of rolling a $6$ has nothing to do with what you rolled before, which makes it $\frac16$.
But what is "memory"? This is what you previously affects what you get next. For example, I have a standard pack of cards. I shuffle them and pick a card, which happens to be the 7 of Spades. What is my chances of getting another 7? That would be $\frac{4-1}{52-1}=\frac3{51}=\frac1{17}$. The chances of getting an Ace are $\frac4{52-1}$, or $\frac4{51}$. As you see, getting a 7 of Spades affects what I get next. But when rolling a die, the probabilities do not change. That is why you will have a $\frac16$ chance no matter what.