Settling on floodplains will provide more food = growth than plains though, and then plains tile can be used either for watermill or workshop for decent production.
That's untrue. Flood plains are a feature like forests or jungle, hence settling on one will remove it and only leave desert in place. Flood plains are as a matter of fact just about the last tiles you would ever want to settle on. Also forget watermills and workshops, this is his capital and apart from the Wheat and the Marble will only get cottages. Production will be low, yes, but it's gonna be very good in commerce. Any production he does need he can get with the whip.
In this turn I moved warrior into city to defend it. I started research of bronze working so I can chop trees with workers and train axemen for war. Totally pointless to research masonry at this point of time becouse marble resource is in the woods. Workboat will be ready in 9 turns and population will grow in 10 turns (if I remember numbers right). My intention is to start work on barracks for the 1 turn before population growth and turn after that start with a worker. Gonna do couple mines for better production and hopefully one of the hills has copper.
What are you defending against? This isn't Deity, you can rest assured neither your neighbors nor barbarians will attack Rome within the next 50 turns or so, so you can just use the warrior for exploration.
Why would you want to rush with axes when you are playing Rome of all civs?

Research some basic worker techs and then head for Iron Working, you have one of the most overpowered unique units in the game, no reason not to use it.
I have a feeling this continent is way too crowded and I could set up only 2 additional cities before borders of others come in the way. So what after barracks and bronze working? Straight to iron working for praetorians with their 8 strenght? Or catapults first? Which one is better suited for early war, or can I start it already with axemen? I know I can use axemen to deny important resources from enemies, like copper, and even capture cities that don't yet have culture defense bonus. But what about capitals that could have archer or 2 for defense with culture defense bonus or even city walls?
Two cities of your own are fine if you are going for Legions*, which you should if you are playing Rome.
Consider researching Pottery for the Granary, which is the most important building in the game, maybe even Sailing for Lighthouse and Writing for Library, though those can be delayed until after Iron Working depending on how quickly you want to go to war. I can not tell you the optimal military strategy yet because you have absolutely no intel. Thus first of all you need to explore and scout out your neighbors and then attack them based on the information you gather. I guess what might work best is if you captured all their lightly defended cities in a first wave with Legions alone, make peace, and then finish them off once you have catapults.
Also you might want to train some warriors before building a Barracks to use them to scout, and later on as garrisons for safe inland cities or extra happiness with Hereditary Rule.
Can't believe I didn't know that was even possible. Feeling pretty dumb right now. [emoji54]
To be fair, a fort only allows passage for your own units, while if you found a city also units of civs that have open borders with you can pass as well.
*I know the game is calling them Praetorians, but they are Legions for all intents and purposes.