a) Yes, if Postgres is exposed to users for some reason. Your changes to Postgres are a derivative work of the original, and thus covered under the AGPL.
b) No. Your application is not a derivative work of Postgres, any more than it's a derivative work of whatever OS you're using.
The license does not include the phrase "connect directly". It says if you modify the Program, your modified version must prominently offer all users interacting with it remotely through a computer network [..] an opportunity to receive the Corresponding Source of your version
"interacting with it remotely through a computer network" may include several hops, the text is unclear at this point. And there is precedent for this interpretation, Microsoft uses the same concept (as "multiplexing") in their SQL Server license terms: https://download.microsoft.com/download/3/D/4/3D42BDC2-6725-... (pdf, from [1], should the document link change in the future).
Actually the GPL has special wording to exclude system libraries and headers from the "derivative work" - since the AGPL does not have something similar (why would it?) one would assume that the derivative work of Postgres includes your app, even if you only communicate with Postgres via http requests going through three levels of indirection
a) Yes, if Postgres is exposed to users for some reason. Your changes to Postgres are a derivative work of the original, and thus covered under the AGPL.
b) No. Your application is not a derivative work of Postgres, any more than it's a derivative work of whatever OS you're using.