Uncaught exception thrown in shutdown
I just tried to setup a fresh Drupal 7.44 with Postgresql and I haven't encountered any issues. The setup went fine. Here are the steps I did.


$ su root - Switch to root user. 
$ su - postgres - Switch to Postgres user. Note that you need to be root first before doing this.
$ createuser --pwprompt --encrypted --no-createrole --no-createdb gerald - Create a new user called gerald.
$ createdb --encoding=UTF8 --owner=gerald drupal - Create a new database called drupal.
Run the installation and select the Postgresql type. 


Another way to determine the culprit is by reading the logs.

For Postgresql

$ sudo tail -f /var/log/postgresql/postgresql-9.3-main.log. In you case it should be $ sudo tail -f /var/log/postgresql/postgresql-9.4-main.log.

For PHP (if you're using Apache)

$ sudo tail -f /var/log/apache2/error.log

Additional Troubleshooting

Also check if there is a problem with timeout in PHP. You may want to increase to default timeout settings.

My stack:

gerald@von-viel ~/ $ psql --version
psql (PostgreSQL) 9.3.13
gerald@von-viel ~/ $ php --version
PHP 5.5.9-1ubuntu4.17 (cli) (built: May 19 2016 19:05:57)


Note: I am using the default configuration of Postgresql.

Hope this helps!