Start, stop, configure PostgreSQL through Mac terminal
The first error indcates the server is already running.  Have you tried connecting with psql or run the following to establish that?

pg_ctl -D /usr/local/cellar/postgresql/9.5.2/data status


If you know it is down, then for the second error with the logfile on startup, either reissue the command with an absolute path to where you can write a file:

pg_ctl -D /usr/local/cellar/postgresql/9.5.2/data -l /tmp/logfile start


Or if that doesn't work for you, try starting without the -l switch:

pg_ctl -D /usr/local/cellar/postgresql/9.5.2/data start


Without the switch, you still get the PostgreSQL logs in the pg_log directory under the $PGDATA - in your case /usr/local/cellar/postgresql/9.5.2/data/pg_log