Using CRM to manage walking groups
With some customization/development, all your wishes can be met by CiviCRM

&gt; Integrated into existing wordpress website 
&gt; Event registration (free and pay capabilities)


Out of the box functionality. Your WP site server needs to good enough to run CiviCRM though. And events are easy to use (free and paid).

&gt; Ability to send out text and email notifications (walking group leader would most likely do this so phone and email information would need to be private)
&gt; Security and privacy of participants (similar to the Remind app) 


Some customization will be needed.

&gt; Ability to set up social network and log walks into site


Please elaborate on what you mean by set up social networks. Walks can be events or activities, you will need to figure out what works for you

&gt; Promote competition with badges/rewards


There are probably a few different ways to do this, may require some development depending on what you need. For example voting on answers like on StackExchange is not available in CiviCRM and will require a reasonable amount of effort.

You should consider working with a CiviCRM Partner/Contributor.