It is always possible to link between facility inventory data, provider data and client (exit and observation) data using facility id and provider id. Facility id variable should be included in facility, provider and client data files and provider id should be included in provider and client data files.

Let's say if you are interested in providers and want to merge some facility variables into the provide file. This is going to be a many-to-one merging because multiple providers are associated with one facility
See below the Stata program for this type of merging.

use providerdata.dta, clear /* you need to use the path where your data files are stored*/
merge m:1 facilityid using facilitydata.dta  /* variable "facilityid" is the facility identifier(it may be a different name in ESPA files); it has to be the same variable name in facility and provider data. Otherwise you need to rename it before merging*/
drop if _merge!=3 /*drop unmatched cases*/
save mergedfile.dta, replace /*save merged file */

The program for merging facility and client data would be similar as above, just replace the provider dataset with the client dataset.

If you are interested to merge some provider variables into the client file, the identifiers would be two variables: facilityid and providerid

use clientfile.dta, clear
merge m:1 facilityid providerid using providerdata.dta  /*as above, both variables should have the same name in provider and client data, otherwise, rename them before merging */
drop if _merge!=3 
save mergedfile.dta, replace

Let us know if you have other questions.

1) It is strange that provider id variable is not included in provider data and client data. It may have a different name and usually is labeled as "provider serial number".

2) yes, if you are merging provider data into facility data, you are dealing with one to many merging. In this cases, facility dataset is the master file and provider dataset is the using file.

3) Without access to the ESPA data, it is difficult to figure out the problem.
Please note when you merge provider data into facility data, the number of cases(facilities) in your master file (facility dataset) will proliferate because multiple provider records are associated with one facility. Maybe this causes the nth value no longer the same?

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.