Hi guys, thanks for the replies! I restarted the map (still Babylon), and tried to anticipate the situation. Founded 3 cities at the start, then I was boxed in (again) and cleared 2 barb camps to the south. After that everything was settled and blocked again. But I think it's alright. There's a nice post on reddit about this "issue". (google reddit old world "AI getting a head start".) It's actually WAD. Dev says, it's a main feature of the game. There was no "empty land" during any time in human history. It's either civilizations (Persia and the Assyrians; Egypt, Greece, Rome) or Barbs (with some of them really strong). So you gotta figure out how to fight your way out of that situation.
I'm currently in turn 140 ... Have 3 main cities, 3 peripheral cities, and one "colony" used for chopping wood mainly. Tech wise I'm doing alright, Babylon is a civilized advanced kingdom with a couple of wonders, and – most importantly – I was able to keep peace with everyone since turn 1. And I might win a war or simply win by fulfilling my objectives, whenever I want to (I hope). – Only problem is the lag/slowdown/memory leak after turn 70 ...
Anyway, the main difference compared to the 1st two runs was that I did not start to scout a lot until I had my three cities settled. I blocked the spots with my 1st and 2nd scout, build 2 settlers ASAP and just tried to settle SOMEWHERE. After that I started scouting the immediate surroundings and realized that I have to take barb camps to expand. So I built a task force and a settler and took those spots. And only THEN did I sent my scouts on a sightseeing tour to Africa, Gallia, Rome and the Black Sea. – This was a nice change from "hunting for goody huts while pumping out settlers (or an invasion force)" as I was used to from other games
And now, I'm just hoping for them to fix the memory leak issue somehow ...