All-to-all modeling for structured dataset?

I have a structured dataset with rows as different samples and columns as different attributes of the samples.

Interestingly, the attributes are highly inter-correlated (i.e. a complex system). I want to understand the system by training many classifier mdoels, with each model taking a column as the target and all the other columns as the features (which here I call such modeling all-to-all). Because the attributes and targets are highly correlated, many models should perform at reasonable accuracies.

Before actually doing that,

1. I wonder if such all-to-all modeling practice exists?

2. If it does, do you know what are the common practice/ state-of-the-art methods in doing such all-to-all modeling?

3. Do we have a simpler method that can model the whole system without training numerous discrete models?