Fix database connections leakage in core (web server)

Right now we connect to the database without any connection pooling. This means that for each new request we will create a new connection to the database. It leads to several caveats:

- Database connection is costly, so creating new one takes some time;
- We close most of them, but sometimes the connection is left hanging in idle or in idle in transaction state, so without some tweaking on database side we may exhaust our connections to the database;

We need to use a proper pooling (either sqlalchemy one or setup pgbouncer).

AC:

- No leaking connections to the database;
- Pooling is used instead of creating new connection each time;