Cute kid. I hope you encourage him to ask these questions and to thing how one might answer them.
Preamble: I think one of the most common misconceptions of mathematics is variations of the expression "the interesting thing about pi is that in goes on forever and never repeats". This is not unique to pi and it is not in the least bit unusual. Irrational numbers are actually far more common than rationals. I don't think most people, not even most mathematicians, intuitively realize just what numbers with infinite decimal expansions mean.
Think about this: For any sequence of numbers you can imagine in any way or order, you can make an irrational number from that. And consider this: between the number .4446 and .4447 you have to "go through" an infinite set, every set, of possible combination of infinite and finite, but mostly infinite, sequences of numbers starting with 4 4 4 6..... 44461284749487493... is in there, 44463141592653.... is in there, 44467777777777.... is in there. 444612345678910111213141516171819202122232425262728293031323334353637.... is even in there. They are all in there. Are you beginning to see how many and common yet how strange and huge these are?
So, there's actually no reason we should be talking about pi. We could be talking about any irrational number and we can make up any possible irrational number we like.
1) Normalcy, patterns and repeating: "If a number repeats it's rational, otherwise it's irrational". Sort of. A rational number will either terminate (reach a 0 and have 0s forever) or reach a point where it will repeat a single pattern forever. Example: 1/7 = 0.14285714285714285714285714285714... which repeats 142857 over and over again. An irrational number can repeat a pattern a few times and then quit. It can even repeat a pattern forever if there are variations and breaks in the pattern. .123012300123000123000012300000123000000.... "repeats" but there are variations so it doesn't repeat the same thing exactly so it is irrational.
Now a "normal" irrational number, one we pick arbitrarily, "shouldn't" have any discernible pattern but have digits in a normal arbitrary distribution. (There are infinite numbers that don't but there are "more" that do.) pi is probably normal but we don't actually know.
2) Specific strings: All things being arbitrary, we expect any particular string n-digits long to pop up once every $10^n$ places we look. That's not very often but as an irrational has an infinite span we expect it to show up time and time again. But we don't expect it to show up in one specific place.
So we do expect the first thousand digits of pi to show up later in pi but we do not expect it to appear at *exactly the 1001 place. If we pick irrationals randomly we'd expect 1 out of 10 will start with the first number repeated twice. We'd expect 1 in 100 to start with the first two digits repeated twice. 1 in 1000 for the first three. And 1 in a googol to repeat the first 100 digits twice.
BUT we know such numbers do exist and as all pattern happen we can make one up. The does exist a number starting with 3.1415926 and continuing with the first 1000 digits of pi and then immediately repeating them. But that number is not pi. (It's within 1000 digits of pi so its close.*)
3) Finding pi in pi: Well, when we say any number will appear in pi we are usually implying any finite number will appear in pi. It's logistically, well, meaningless, to find an infinite sequence within a sequence because ... well, if the inserted sequence is infinite it has a start but no end, and if it has no end we can't put anything on "the other side", and therefore we aren't actually "inserting" it.
But, as we can make any sequence of numbers into an irrational number, we can make some sort of "fractal" decimal where patterns of 31415926 are inserted inside themselves in large and small and telescoping patterns. (But that sure as heck is not a normal irrational). Exactly how the pattern is defined is up to us but as the number has no end we can't expect it to be symmetrically nested or anything like that.
*[Notice I just implied there are 10000000000 googal different numbers all within a thousand decimal places of pi! That's pretty much my primary point. There are a lot of irrational numbers and they are very tightly packed and the are infinitely varied.]