Hi Rakesh -
Locating this info in the various logs is likely to be more trouble than it is worth. You can see a list of the server logs, location, and content here: Server Log File Locations
Having said that, the best method for acquiring the information you require is by accessing the PostgreSQL repository, and it doesn't require you to sign-in to server. Once you set-up access to the repository via the tabadmin dbpass command, you can approach it one of several ways: you can utilize the sample performance workbook, modifying it to your needs or write your own query against the repository. Once that is in place, you can simply use Tableau Desktop to view your output and/or schedule dashboard to run on a regular schedule.
See here for additional info:
It may not address all your questions but definitely worth a look would be the LogShark tool.
This tool basically spits out a number of workbooks and one of them is all about Backgrounder. It's very handy in that it summarizes overall errors but you can see a breakdown of both your extracts and Subscriptions and this is all done from existing log files.