I have yet to research Agriculture. Should I switch from Writing, delaying Libraries? IMO yes, since Libraries provide increased beakers, which are downstream from commerce. Nevertheless, city expansion should begin ASAP.
I also have yet to switch to Slavery.
I didn't notice you lacked Agriculture but, looking back at your start, it totally makes sense. So, no worries.
You don't need it right now (Workers by Opporto can part cottage instead of part farm, if you want. You can research Agriculture in 3 turns and there's ample time to complete Writing + Agri before the next settler is out from Seoul.
That said, it would have been ideal to research Agri before Pottery (and instead of Sailing) to benefit from the discount.
Worker by Lisbon can chop riverside instead of farming. There's also that.
Should you switch to Agri right now ? Perhaps, yes. There are farms you want to start asap (Lisbon) and the Library in Opporto will not complete anytime soon. So, Writing is not a huge priority.
You will probably need a whip to complete your first Library.
So, you will need to switch into Slavery at some point to do that. When ? That is uncertain. It's best to do it when workers are done / settlers are in transition. The longer you wait, the more "1 turn" gives you ; so it is generally better to switch earlier (less loss).
re : horse archers (or military in general) :
It's good to consider the strengths/weaknesses of your top unit but, as hinted by Jorissimo, the outcome of combat has a lot to do with Production capabilities and Unit mass.
If you have 2-3 times the number of cities the AIs have and twice their numbers, it doesn't matter if some of their units counter your own
re : research and gold :
In general, keeping gold "in reserve" is not very useful. Why keep 120 gold and research at equilibrium ?
There are techs you want fast. You should use your bank to get them

Also, about the slider, the absolute numbers (+25 science, -8 gold) are a lot more important than the %. You could have a low % and huge output, for example.