(Assuming you remove all terms which would result in 1/0…)
Well, it trivially won’t converge if pi is normal, because then there will be an infinite number of 1/1 terms, so it will keep trending upwards indefinitely.
However, we don’t know if pi is normal.
On the gripping hand, pi being normal is a sufficient but not necessary condition for non-convergence: It is very easy to construct non-normally-distributed infinite sequences which won’t converge if added in the way you propose.
So I don’t see any special reason your sum would converge, and since there’s so many ways for it to diverge, I’m pretty sure it’s going to be divergent.
No. A series 1/9 + 1/9 + 1/9 + … will diverge if it has an infinite number of non-zero terms, so yours will also. And the decimal expansion of pi contains an infinite number of non-zeros; otherwise it would eventually be …0000… forever which happens only for rational numbers.