How is a 3rd prince younger than Wallace who is the 100th or so?
While the lore reason is because Amagi is treated as a sex bot, the main reason we missed the sex scenes is because the author is allergic towards showing Liam's relationship with Amagi or any woman. They would rather focus on the gay jokes.
I am an LN reader, but basically the ranking is done by "order of succession" as set by the emperor himself rather than by age.
The 3rd prince has a fairly ... tragic background and terrible upbringing to say the least. I'll shortly summarize it thusly:
Prince Cleo is in his current role because the emperor simply (and only) wanted to see him struggle and suffer. Cleo's mother also did some extreme things to make it happen.
We'll also see some of Cleo's older siblings from the same mother later in the story as they are introduced.