I can't answer the first two questions in a definitive manner. However the rest I can handle:
How would the game work (i saw somewhere else it's RTS style)?
It's not quite RTS, but it's not turn-based either: every human player takes their turn at the same time. Once all players click "next turn", the AI civilizations, city states and barbarians all take their turns in order. Then the human players' next turn begins.
This causes some big issues when interacting with other players, and is a big griping point for many. For example, wars became a battle of who can click first - if I can blow up your unit before they get a chance to attack, then I'm in a hugely advantage position. It's unfortunate to the point where a lot of us don't even both with multiplayer (except cooperatively with friends) because when everyone moves at the same time, the game loses a lot of the carefully planned strategy you get in single player.
It's worth noting that once you click "next turn", you can't do anything even if other players are still moving - so it's to your advantage to delay the end of your turn a bit in case something comes up.
How long will the game take to complete?
Depends highly on the number of players and the game speed, but significantly longer than a single player game of the same settings. Some turns you may be blazing through waiting for construction while another person is taking his sweet time waging war, and vice versa. Even playing on quick/small map can take hours. Set aside a large chunk of time.
Can I take a bathroom break without abandoning my civ temporarily?
Yes, you can just sit on the "next turn" button and not click it while you do something else. You would just hold up everyone else from getting to the next turn.
Can I chat to the other players?
Yes. There is a chat box that appears in the top right corner of the screen.
Best Answer
From your screenshot, it appears your opponent has built the Great Wall. You're crossing into his territory, and the effect of the Wall is that your movement will be penalized.
I'd be willing to bet that if you hovered over the hex just left of the enemy unit, your Elephant unit would path down and to the left towards it, rather than down and to the right, which would cross over into enemy territory and take a movement penalty.
The Great Wall is an excellent choice for a Civ who is planning for war, as it gives attacking units a serious disadvantage - marching on a city in this case is going to be a slow process, and it will give the defending city's siege and range units extra turns to decimate the attacking forces.