Am I supposed to? My play style is to enter into eras around the estimated year that they start IRL. It gives me a sense of scale and immersion in my games. If I am way behind research it feels rushed. If I'm lagging behind it's a problem for me.
No of course, you are supposed to play in a way you enjoy. However i always thought Civ did a poor job of respecting the time scale (and Civ6 is probably the worst so far) so if you purposefully slow you down to respect historical timing, you are handicapping yourself. Again, nothing wrong with this, but keep it in mind, if you ever play on higher difficulties you'll have troubles when the AI hits Industrial 100 AC and you are still in Classical. I actually never look at the date myself and simply consider game turns. I think it's perfectly possible to win by Culture with Greece between T250 and T300 even for someone who's not too good at fast finish like myself.
I have several artistic GP lying around in my game, waiting to fill Great Work slots. They just need buildings to have slots.
OK, didn't really look closely for this. It's normal you get Musicians before you have slots if you don't get one of the early wonders with music slots. For Writers, it will be better when you play a post-patch game as Amphitheaters now have 2 slots. I don't like having too many Art Museums, 1-2 when playing culture and then Archeological. I find them easier to fill and theme. Theming is important as it doubles your culture and tourism output and art is pain to theme especially with the way trade works.
I should have thought of that when I settled my first cities early game. Correct me if I'm wrong, but overlapping bonus happens when two cities share the district's yields + bonus yield for both cities because of the overlapping?
Yes. Both Factories and Power Plants affect cities in a 6 tiles radius. On most maps, a well placed Industrial Zone can easily cover 4 cities. you can get more with perfect placement (not always possible due to map). On your map i would say you could probably get 3 cities covered by well placed IZ. Factory + Power Plant gives you +7 production, so if you cover 4 cities you actually get +28 production, making the time invested in building them pay off much faster.
And as you said, overlap is when a single city benefits from several IZ. You can get some really nice boosts if a city gets the benefit from 4-5 IZ.
Zoos and Stadiums from the Entertainment District also have an area of effect.
I did realize that while I was playing mid-game. :/
Some very wise men said "learning is making mistakes". That government might have it's uses in a few situations, but i still have to find those.
I have a confession to make though (I'm planning to post a full thread on this in the near future, but I think it's relevant to talk about this for this thread)
No need to be embarrassed. The point of playing a video game is to have fun and not everyone find it fun to fight against an AI that start with 3 cities and reaches Industrial in the late BC

Some tips are of course more relevant to the higher difficulties that most people in the strategy sub-forum play on.
Do you know how adjacency bonuses work for farms and districts? It would be a good start to improve your games as you can really get more food and production just by placing your stuff in the correct position.