I'm wondering if building more steel factories would be beneficial or just cause problems in other areas such as population, space, culture, etc?
It might cause problems, mainly with population (and space, if you're not going to buy expansions with diamonds - it's just a temporary problem though). Don't worry about culture, your fellowship and neighbors will help you gain some bonuses with neighborly help later on, even if you keep it at zero.
Number depends on your playstyle (basically on a "free" time you have):
Nr.1: build as much as possible, seriously, you might end up with 15-20 manufactories at that will be just ok;
Pros:
a) drastically speeds up your progress in tech tree (because you'll be able to spend the surplus to get more knowledge points);
b) you'll always have necessary goods for upgrades, tournaments and negotiation in provinces;
c) your fellowship will love you;
d) you'll climb up ranks (gain score) really fast.
Cons:
a) you'll have to forget about military aspect of the game - that means no armories (with such strategy armories will just take your precious space, culture and population);
b) you'll have to cut the number down eventually - you'll run out of population and culture;
c) your fellowship might end up hating you - because you'll take all the trades (lol);
d) you'll need to log in several times per day (at least every 3h up to 3-4 times) to reset the production and gather supplies - otherwise effectivity of such strategy drops;
e) your city will look weird.
Nr.2: build 8-10
Pros:
a) you'll have space for upgraded barracks and 1-2 armories which means that you'll have an army too;
b) your progress in tech tree will be going at normal rate;
c) it's a "normal" game experience, because you're mixing both combat and trading;
d) most of the time you'll have necessary goods for upgrades, tournaments and negotiation in provinces;
Cons:
No cons.
Nr.3: build 5-6 (I strongly advise you not to build less than 5)
Pros:
a) you'll have a lot of space and you won't struggle with population;
b) you'll never have to cut down this number at any stage of the game;
Cons:
a) you'll struggle with goods for upgrades, tournaments and negotiation in provinces (most of the time);
b) like with strategy Nr.1 you'll need to log in constantly to reset the production of goods;
c) your progression in tech tree will be super slow.
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Personal recommendation:
1) build at least 8 steel manufactories (up to 10), at least 4 crystal manufactories (up to 5) and at least 3 elixir manufactories (up to 4);
aaaand some tips:
2) don't build armories at all or build just one (at some stage of the game quest for building/upgrading multiple armories will appear - just decline it);
3) don't waste diamonds (if you're going to get them) on cultural buildings... magic residences and workshops are bugged right now, so don't buy them either for now.
P.S. I really regret that I didn't know about strategy Nr.1 when I started....because I can play this game all day long due to my work, but it's kinda "nerd-y" and is not suited for everyone....so just pick whatever works for you.