Yes, but only if your server can handle a parameter that is there, but is empty, as in:
www.myserver.com/update?t=&h=39&w=-48
To do this, you would use the query section in the webhook configuration to set each of the parameters (t, h, w, etc.).
Then, in your Particle code, you would create a JSON object but only include the valid values.
The result values from the server are returned in a hook-response event, so yes, you can check the response from the device.
My tutorial: