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
You'd end up coding an install profile (PHP) that directly answers all config questions (site name, db name, db user, db password, and so on). It's not hard, but there's already Drush that does this very well so you could consider triggering Drush in the background (using it's "site-install" command), then there'd be no progress bar from Drupal (but instead your own custom form/page with a progress indicator until the server [Drush] finishes preparing the new site/domain/folder.
Personally, I'd use Drush and hide it behind an authenticated form. But, if you can't use Drush for this, then look into generating your own settings.php file (with $config variables for everything needed - site name, URL, db name/user/pass, etc.)
Well, simplytest.me does something like that, but it also involves shell scripts and so on (you need to select a contrib module to skip the installer). simplytest.me is open source, so you can look at what it does.
It's not possible to do it without changing the installer I believe, although in Drupal 7, it used to be possible to pass the language and profile to it through the URL and if you had the database settings already in settings.php, it started the batch immediately, that no longer works in D8 but could possibly be restored.