API Calls - Thread Activity

What exactly are you hoping to accomplish with your API calls? The calls above are both for creating new threads or replies using data from another source, but it sounds like you are hoping to do the opposite: get data from Community to add somewhere else? In which case you need the Show endpoints:
https://community.telligent.com/community/10/w/api-documentation/61549/show-forum-thread-rest-endpoint
https://community.telligent.com/community/10/w/api-documentation/61564/show-forum-reply-rest-endpoint

I would need more details on #3 for what you are trying to accomplish, but I might start with the content subscriptions endpoints:
https://community.telligent.com/community/10/w/api-documentation/61503/content-subscriptions-rest-endpoints

If you are hoping to monitor actions in community and transfer data elsewhere, you'll need to set up Webhooks. This is a way for Community to send a call to another URL endpoint so that it can react to the events in an external system. 
https://community.telligent.com/community/10/w/developer-training/53147/webhooks

Thanks for the quick reply!

The goal is to pull threads created or replied to in the community into a personal dashboard within our website. For example if I post a new question in the community it should appear in my dashboard on our website. The same would go for a reply that I post to an existing thread in the community.

#3 pertains to pulling in a list of any thread that I have turned on notifications for. The same behavior as above, if I turn "Reply Notifications ON" then I would want this thread to appear in my dashboard on our website. We are only interested in followed threads.

https://community.telligent.com/community/10/w/api-documentation/61549/show-forum-thread-rest-endpoint
https://community.telligent.com/community/10/w/api-documentation/61564/show-forum-reply-rest-endpoint

If I use one of the calls you listed above, how do I define the specific user associated with the those threads and replies? I see a forum ID called out but not a username field.

You'll want the List endpoints then:
https://community.telligent.com/community/10/w/api-documentation/61546/list-forum-thread-rest-endpoint

https://community.telligent.com/community/10/w/api-documentation/61546/list-forum-thread-rest-endpoint

There are a number of premade endpoints there that might be useful (for instance, the mythreads ones) that would help with #1 and #2.

Unfortunately there is not the capability to list all subscriptions for a user, only a way to list subscriptions for a certain piece of content. It would be possible to pull all threads and check the List Content Subscriptions endpoint for each content to only include content with a certain user Id present. But I'd recommend starting with the above endpoints to see if they fit your needs.

Thanks Steven! I really appreciate your help.