Germany had a government, the old one keeps going until the new one is in place. You'll notice their political system didn't grind to a halt.
Your 2nd point is a great example of why PR would be helpful. At present as you say both parties with a realistic chance of election only tell us what we want to hear. Under PR you can have an unabashed communist party, or the BNP, or Greens and still get seats.
It'd also break up the too broad major parties. The Tories should be at least 2 parties: right wing pro Europe and right wing anti Europe. That way they'd have avoided 40 years of infighting on the subject and if you vote right wing you'd have a choice on if you liked the EU at the ballot too.