RandomLeader
Chieftain
The weird part is that you say that one of your cities is building only military yet you are the one that gets attacked. If you connect horses and copper early enough there usually is a window of opportunity in which the AI defends mostly with archers. Use that to attack, not to defend. You can attack with two early cities if the enemy is close enough. Three or four if not. Unless there's a crazy warmonger you should really be the first one attacking.
That was case related. I only research iron working if I don't get bronze anywhere near. Sometimes just horses will do and I'll make lots of chariots. On some occations I research it if I get goodies that are covered with jungle.
Montezuma isn't too bright every time if he is your neighbour but this time he managed to pull horsearchers after a stack of jaguars(which I destroyed quite easily). Horsearchers eventually started to get the best out of me and he refused to have peace. I couldn't get friend to war and last drop was that he bribed Mansa to declare. I had 3 cities at the time and every city made spears and axes with whipping. That fun ended (in that game I had iron) after 2 rebuild of that resource and I always tried to defend it. I still had chariots thoe. I must say that I had quite good army and I was doing just fine for a "long" time with that war. But somehow I think that monty stopped research after getting horseback riding. On the other hand my research went down. Maybe I should have attacked myself little earlier? Oh.. forgot.. in that paticular game I raized Montys closest city with 4 axes. I had construction but I had to waste most of my cats for collateral damage to few big stacks. Not enourmous but respectable stacks anyway. Mansa came with longbows.
Thats small story and it doesn't have all the details. I'll try to learn how to make a presentation so then it's more easier to point finger

Mysticism is usually a top priority to expand your borders. Chop that Monument to do this as early as possible. Pottery can be good if you have lots of food or are expansive (whip grannaries) or lots of grassland (cottages). If not, head for writing. A library in your capital will really help your research, plus you can most probably run a couple of scientists in it.
I rearly ran into culture problems and usually I get my research going quite well. I don't think that I have too big problem there. I usually build eventually every building to every city. My priorities are granary, courthouse and library. That depends also where I'm going and which city I'm improving but if that is my militarycity I always try to commit some other city to make military if I need to sidetrack with that.
I'd say it's better not to build early wonders if you don't have any bonuses (Industrious or Stone or Marble). You're tying up a city that could build either war units or help you expand (settler) or improve (worker).
You might be onto something here. This might be my biggest flaw. Maybe I'm too narrow minded when I think I need something like pyramids. Biggest problem for me is that I don't like early religion and I sometimes overlook monarchy which basicly means that I need to get happiness from somewhere else (representation is too often my goal). Hmm.. How do I check where AI has builded wonders if I really want to have that? Meaning I would gladly try to conquer that city. Sometimes I do it but it's always accidental.
Why don't you post a game here? We could follow you along and give advice. Before each round tell us what you plan to do first, and we'll tell you if we think there's a better plan.
That will be my first goal. My spouse is little pissed off about that game so and I'm starting my own business. And I got my dayjob still so I got my hands little bit full at the moment. Anyway.. Some people sleeps at night, I think it's overrated. I'll be trying to start a thread within this week so help is really needed. I must confess that even when I first time played prince level it didn't feel so hard. Monarch is still wayyy ahead of me. On the other hand I usually build spaceship with prince but sometimes its more like playing for fun in the end (stopping the rocket and looking how far future techs I can go or making tons of money and having fun by bribing other civs to WWIII).
Edit: to the beginning. All my cities are not making military before I'm at war. I always have only one city that builds only military. Little language barrier from me I think.
EDIT EDIT: btw, it's hard to say what my biggest flaw is when people point out them from left and right. And that is a good thing cause sometimes one can be so close to a problem that he turns blind

EDIT EDIT EDIT: and the game is still warlords.