Does The people's hammer need to deal damage to push? If not, does it need to make range?
It only needs to be able to target them legally. If it does after the damage (or lack of damage) resolves push them.
Is push mandatory? If I hit someone with Proletariat, do I HAVE to push them or can I decide not to?
As long as you are in melee range, regardless of whether or not the dice make range (maybe due to elevation), the push would occur at the end of the attack. That's how I've played it. However, the rulebook says that Push attacks "move a target hit by this attack that number of hexes", so it's very possible that the Push is dependant on damage taken if you read this literally.
I believe that Push is mandatory. It does not say "up to" and the glossary says it's a "forced" movement "that number of hexes".
Maybe Paul will chime in on this. I'd be interested in his answer.
During one of the tournaments, they ruled it was "up to", but that might have been a mistake. 
Well, you can make any push beyond 1 end on the original space if you move people the right way
Official Rule Time!
1) Push is not optional, and must be the number specified in the ability; i.e., "Push 3" must always Push the target 3 hexes, not 1 or 2. However, as Humble-Knight mentioned, you can make any Push beyond 1 end on the original space if you move people the right way.
2) Push is separate from damage. If Proletariat attacks a target and deals no damage, that target is still Pushed.
3) Characters are immune to the effects of their attacks. This includes damage, which is the most common case, as well as effects like Push. TL;DR - Proletariat cannot Push Proletariat.
Thanks, Paul!
Thanks, Paul
Shiny - many thanks, Paul! 