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Some horses allow me to enter in low-level competition but some don't even if they haven't won 20 comps. Is it because of skill level / age?
Competition
There's a section in the breeders manual about that:
https://au.howrse.com/aide/manuelEleveur/?type=section-3-3#anchor4
Yes, most certainly it'll be because of the skills. As it says, "This value is recalculated every night." - it's some % of the best horse ingame.