Per the Breeder's Manual, low-level competitions are restricted to horses which have less than 20 wins, and, as of today, no skills over 3300, and no skills with a GP over 4620. Because those skills are recalculated every night, they may look slightly different tomorrow. You can read more about it here.

As for your EC comps, it's hard to say why you can't enter your own horses without links to said horses. It's worth noting that unless you have the VIP perk, your comps can only run once a day--perhaps double check to make sure they haven't run already that day? You've also reserved most of them for low-level horses--there are much fewer of those on the game than not, so you're likely to see less participants--I would also make a guess that perhaps your horses' skill levels are too high to find those comps.