Sudden loss of air on tubless tyre, and now it won't inflate
Your valve should NOT be depressible into the rim.  You should have a presta ring screwed in it, else nothing is retaining the valve and your air probably leaks from there!

If its not the case, if the tyre is seated correctly on your rim, inflating it with a pump should be easy.  Maybe you have a dent on your rim where your air is leaking.  

Yes absolutely, you can get a new valve for tubeless tire.

A good way to find where your air is leaking from is spraying soapy water around and check if an area makes bubbles.  It will tell you exaclty where your leak is from. 

Cheers