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I have a panel dataset and to check the robustness of the results, I'm re-estimating the models with each panel unit excluded once. Does this robustness check habe a particular name?

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Depending on the context this is sometimes called Jackknife resampling

In statistics, the jackknife is a resampling technique especially useful for variance and bias estimation. The jackknife pre-dates other common resampling methods such as the bootstrap. The jackknife estimator of a parameter is found by systematically leaving out each observation from a dataset and calculating the estimate and then finding the average of these calculations. Given a sample of size n $n$, the jackknife estimate is found by aggregating the estimates of each $( n − 1 )$-sized sub-sample.

It may also be referred to as leave-one-out cross validation, which isn't exactly the same but very close.

Leave-one-out cross validation is K-fold cross validation taken to its logical extreme, with K equal to N, the number of data points in the set. That means that N separate times, the function approximator is trained on all the data except for one point and a prediction is made for that point.

Well, that's the name for leaving out a single observation at a time. Strictly speaking, it isn't the name for leaving out each panel unit once. When you resample in way that accounts for the panel or time series structure, that is usually called Block Bootstrapping, in this case a kind of cluster bootstrapping.

Personally, I would call this block-jacknife or leave-one-block-out cross validation.

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Outlier check. You only need the outliers not all elements.

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