I am trying to contribute Drupal modules and placed my first project on drupal.org.
Is it possible to see if a proprietary project on drupal.org has been downloaded and how many times?
Have a look at the "Usage statistics" page of any module on drupal.org, which includes all sorts o relevant information.
As an example, checkout the Rules module, for which you can find those statistics at https://www.drupal.org/project/usage/rules . Apart from a nice chart, it also gives you the statistics by Drupal Release (D6, D7, D8), and this for each Drupal Release for which the module does have some release. If you scroll to the bottom of those statistics, you even get more details (for the last 5 weeks) of every (minor) release of the module.
Attention: don't confuse "using a module" with "number of dowloads of a module"! E.g, for the Rules module, there are currently 296,819 sites reporting that they are using this module, while there have been 2,642,716 downloads so far. Open question: does that mean that only (about) 10% of sites for which this module was downloaded actually still use this module? Answer: I don't think so ...
PS: Replace "rules" in the above URL by any other module's name to find similar statistics.