The same question goes for using the shapeshifters activated ability to discard a Phyrexian dreadnaught from my hand.
It's the VS's appearance on the battlefield that will determine if the ability triggers.
Static abilities are always operational when the object with the ability is in the appropriate zone, so there is not a moment the VS is on the battlefield that it's not a PD, so its triggered ability will trigger.
An object enters the battlefield when it is placed in the battlefield from another zone.
The object in this case, the VS-become-PD, has not changed zones. All that happened is that its characteristics (name, type, text, etc) changed. As such, it has not entered the battlefield.
For the first question, yes, you will have to either sacrifice the Shapeshifter or Sacrifice other creatures. The Shapeshifter's static ability applies as it is entering the battlefield, so it is just like a Dreadnought is entering the battlefield.
For the second question, the answer is no. The rulings of the Volrath Shapeshifter say:
The creature it becomes is not "entering the battlefield" because it is already on the battlefield.