Are all relationships between categories, brands and products maintained while migrating?

Yes. StagingPro keeps the structure of the data together in such a way that relationships between objects are maintained even though the indexes in the actual store change.

StagingPro is essentially creating an additional store within BigCommerce and recreating the original environments data into that new store. Since, these are ‘new’ items as far as the new store is concerned, BigCommerce will change the indexes, however, StagingPro identifies these new indexes and ensures that all data is put back in the right place.

You can find a list of dependencies here as well as a list of known data migration issues.