As to your question of which NWs are mandatory vs which are optional
The mandatories:
National Epic - place this in a food heavy city and run tons of specialists under Caste System. Great for producing GPeople which become more important the higher level you go. Great synergy with Globe Theatre
Globe Theatre - Great synergy with Nation Epic, no unhappiness in a city that will have
problems, means you can run more specialists and increase your GPP output.
Heroic Epic - Faster military unit production. I like this in a city that I believe will be my best
city for fast military production. Good synergy with West Point.
West Point - If the game goes this late. I usually build this in my HE or Ironworks city. Extra nice in a coastal city if you need a lot of naval support since in naval engagements promotions really help.
Ironworks - This goes in second best
city, preferably with access to coastal and land based health resources (granary and harbor)
Forbidden Palace - if your empire gets big enough, you will want this, period.
Oxford - a must, goes in the city that will have the most base
along with all science multiplier buildings.
Wall Street - A must, goes in the city with second highest base
and all gold multiplier buildings. Sometimes I will put this in the same city as Oxford, but only if I absolutely have to due to no other
rich cities. Good to put this in a shrined city if you have one.
Optionals
Moai Statues If I build it, I put this in a coastal city that has a lot of water and very little land tiles. It can turn that city into a half way decent naval yard. Decent synergy with Ironworks and possibly Westpoint. Situational to the map.
Hermitage I usually don't build this unless I have a neighbor like Zara and need some major cultural pressure to keep good tiles, or if going for a culture victory. One decent use of this, is coupled with a few culture buildings, it can sometimes help you extend your borders right to an enemy's city(s) allowing you to quickly conquer them on the turn you declare. Synergizes well with the Mandir, Academy Cathedral, etc building type.
Mt. Rushmore - Comes late, so I don't often build it. If you know you will be in prolonged modern war, it is really nice to stave off war weariness.
National Park - I have never built this. The power of chopping trees is just too great. If you can leave your GP city well forrested, it can synergize nicely with National Epic or possibly with Globe Theatre.
Red Cross - I build this if the game gets this far and I know I will war. A free medic 1 promo to all units built there is fine and if you build it where you build troops, you have lots of
and it is done quick. Good synergy with Heroic Epic, West Point (could get troops to Medic 3 easily) and Ironworks.