I read through these forums and saw that some people were saying don't build too many city improvements due to the maintenance. As I have been using that advice and building them only when really necessary, I saw that half of my cities were producing wealth since I didn't need to build anymore workers and am researching for better military units. I saw my lux slider is at 40% and civil disorder still happens from time to time.
If you mean: "half of my core cities" then this is an indication you aren't expanding fast enough, build units and conquer!
But If you do expand enough, then its likely that far more than half of your cities are of the totally corrupt kind. In these cities, building wealth is indeed what most people tend to do. Or at least I do.
Is it worth it to build temples/colosseums in those cities?
If they are corrupt cities, then no, it will cost you more gpt than you gain.
If they are core cities, then yes, but only if you
insist on respecting your neighbors right to exist. Otherwise, no, just build more military and go conquer someone, preferably someone with lux resources.
Also, should I build more courthouses or switch to republic to reduce corruption (I am currently in Monarchy).
This is not a matter of either this or that! You switch to republic because its an all around better government (safe for a select few exceptions)
Then you determine if it is worth building a courthouse on a city by city bases.
Usually, republic makes it worth to build a courthouse in more cities.
I hate republics due to their unit support. To add on to that, how do I cope with that if I do switch to republic?
Consider this:
In monarchy, a size 12 cities can support 4 units, with 100% going to science, this city will also produce 13 science beakers (assuming all tiles have a road).
In republic, the same city can support 3 units via its city core. but its produce 13 commerce bonus. So if you set the science slider to 50%. it is
still producing 13 science beakers, but also 13 gold, that can be used to support 6 more units. So the total unit support is 3+6=9 And if you build a market, it can support 3 more units, = 12 units.
So what can possibly be your problem with unit support?
In vanilla and PTW, this is even more extreme, as a republic bonus commerce could support 13 units there, and 19 with a market.
Whats more, you will likely not need that many units per city, so you can increase the science slider and get
more than 13 science beakers.
In monarchy, if you have less than 4 units per city, the unused unit support is wasted.