Yes, your understanding is correct!
Regarding the merging: You can only merge two files at a time.
If the final unit of the analysis is clients, I would merge the client file with the provider file first (with client file as the master file), then use this merged file as the new master file to merge with the facility file.
Regarding the weights: Which weight to use depends on the unit of analysis. If you analyze facilities, for example, to look at facilities characteristics, you would use facility weight. When analyzing clients, you would use the client weight. They are not the same.