What is the point of doing cross validation like this? I have some data and some models that I'd like to cross-validate.
Here is my approach.

*

*Take my data, which has roughly 10.000 rows.

*Generate 10 test sets by simulating, with replacement, 1.000 rows from the original data.

*For each of those 10 sets, fit my models.

*For each of these 10 fitted models, test them on the full, 10.000-row data set.

*Take the MSE for each set, and then average that across all 10 sets. That is then my performance metric.

Is there something wrong with this approach? Does it work? I was reading wikipedia's page on cross-validation and could not find this exact procedure. Does it have a name? If not, what are is weaknesses?
 A: Yes, this method will give an optimistic (read: bad) estimate of your model’s performance. It has what’s alternatively called data leakage or train test bleed. It’s something that well-designed cross validation techniques must not have.

The problem comes in here:

For each of these 10 fitted models, test them on the full, 10.000-row data set.

Your model was already trained on some of this data, and now you’re testing on it. Because you’ve already seen some of these examples, your model will do very well on them. This is not representative of real-world performance (where the samples are unseen).

I also wanted to check what you meant here:

For each of those 10 sets, fit my models.

I assume you’re fitting on all the data except the sampled test set; otherwise you have the same data leakage problem as above.
A: As AryaMcCarthy already explained, including the cases drawn for training into the test set gives optimistically biased (too good to be true) results.
If you'd exclude the training cases, your procedure may be described as a variant of out-of-bootstrap error estimation. Bootstrap because you draw with replacement (cross validation draws without replacement). Variant because out-of-bootstrap usually draws as many samples as there are in the data set. Note that out-of-bootstrap is typically repeated far more often than only 10 times, but there's no hard default on this.
