Share text via HTTP request I'd like to add an item in the system "Share" menu that sends links and other small text via HTTP POST requests.

On my PC i'm using curl + a shellscript.
The share menu simply lists apps that have an appropriate intent filter for the share action with the type of data being shared. So the only way to get a new item on the share menu is to install an app that does the thing you want. If one doesn't already exist, it's not that hard to create one following the official documentation, but since this isn't a programming forum, we can't help you any further.