Have you taken a look at the Invoke-WebRequest commmand? (requires powershell 3.0 or above) I believe the following would work for submitting the data
#POSTing data
Invoke-WebRequest http://www.websiteURI/ `
-UserAgent 'My User Agent' `
-Credential $cred `
-Method Post `
-Body @{
q = 'powershell variable scope'
num = 30
lr = 'lang_en'
}
For your bonus, the result of Invoke-WebRequest contains a collection of the InputFields on the page, which you can use to get a list of form elements to set.
#List input elements
Invoke-WebRequest http://www.websiteURI/ | select -ExpandProperty InputFields