As far as I know there is a number of tools which collect some statistics concerning some PostgreSQL issues (connections, disconnections, queries, the most frequent events etc.). Some of them rely on Postgres Log or use other approaches.
I have a question. Do you know is there such a tool or a set of some actions which can give a precise correct information whether db server is operable or not? As I understand unfortunately not all the high-severity(FATAL or PANIC) messages in Postgres Log are really serious.
It depends on what "operable" means in your context. To answer the basic "is Postgres up and accepting connections?" question, have a look at pg_isready:
The only FATAL I can think of offhand that _may_ be ignorable (depending on your situation), is the FATAL thrown by too many connections. My opinion is that every FATAL, and certainly every PANIC, is grounds for immediate investigation.
