I would like to get a list of geodatabase feature tables from FME. Is it possible with Python? Or another transformer?
I would use a dynamic reader (ie one with just a single source feature type with the merge filter set). Then expose the fme_feature_type format attribute. This will give you the name of the table being read.
You can run the data through a DuplicateRemover too, removing duplicate values for fme_feature_type, in order to get a list of unique table names.
There is a Schema format that returns the schema of a dataset, but I think it works only with file-based data, and wouldn't work with a File Geodatabase (though it might with a Personal Geodatabase).
Yes, you can list object in any ESRI connection; as long as you have the ArcGIS Server installed on that machine you can use the list object in ArcPy to return a array of feature classes.
You will just want to make sure that is you are dynamically checking a variety of Geodatabases that you have your parameter for your workspace set.