How to install Drupal without interaction via web UI? Is it possible to confgirue installation profile so it can be installed via web UI as usual but without any interaction, just the progress bar?
You can Install Drupal using Drush without interacting WEB UI.
You have to use following command 

drush dl drupal-7.x --drupal-project-rename=D7
cd D7
drush site-install standard --account-name=admin --account-pass=admin --db-url=mysql://YourMySQLUser:YourMySQLPassword@localhost/YourMySQLDatabase


Ofcourse You have to install Drush if not already installed and change Mysql Database details in command as per details. 

Choose installation profile that you want to install 

site-install standard OR site-install minimal