Including column of indices as predictor for model?
Including column of indices as predictor for model?

No, don't include indices. They don't provide any meaningful information about the problem. They are just an enumeration. Take them away.

Is there a reason why including this column of indices gets you to 100% F1 score or even accuracy?

You probably get that when evaluating in train set, in test you wont get that. If you do, then you have a problem with your data (it might be sorted).
What does the underlying algorithm do here? Something like this for a decision tree
if index == 1:
 predict =1
if index == 2:
 predict =0
.
.
.

As you can see there is no meaningful information about the problem here and the prediction in train will still be zero.
Conclusion: don't use indexes as a training feature.