Chuckles, to be honest I don't know the answers to all your questions as it was a while ago. I was doing No Quest for the Weary, which as you know has a lot of QP. I had completed the required number of Platinum quests and was just using up the remaining QP, so I was very near the end. I had annexed Advorton to reduce the number of QP needed to do quests, and it made my game run on a lot longer because it seemed like it was taking forever to use up all those QP. So I really don't know what the renown was, but it must have been very high. All the buildings were in place. The character I replaced the monarch with was a new monarch I created. I don't understand why I was able to get the Baby Monarch achievement when others haven't - the only thing I can think of is that all those quests had raised my renown to a high enough level. Next time I start the game, I'll go back and check how much renown that Ambition ended up with. It will be a little more than what I had when I brought in the new monarch, but not too much more.
Edit: After reading about the trouble other people are having, now I'm wondering if I actually did get this achievement or just thought I did! Am I ever going to feel silly if I didn't really get it. I hope I haven't led people astray. Going off to check ...
Ok, I've just checked and I was wrong. I did not get the Baby Monarch achievement. My mistake. Sorry about that, everyone. For whatever it's worth, I finished that Ambition with 960 Renown. Maybe the cutoff is 1000?