So long as neither the caster not any of the other 9 creatures is large (or larger) Tiny Hut doesn't fail:
Nine creatures of Medium size or smaller can fit inside the dome with you. The spell fails if its area includes a larger creature or more than nine creatures.
However, Find Steed doesn't allow you to "copy" spells. It allows you to share the effect of one spell between you and your steed. It's a subtle but important distinction. The wording of Find Steed you're alluding to reads:
While mounted on your steed, you can make any spell you cast that targets only you also target your steed.
This duplicates many effects (if you make yourself invisible, your steed would also become invisible) but, it also means that certain spells function differently.
In the case of Tiny Hut, the the mount would count as "you" with regard to the following clause:
The spell ends if you leave its area.
Meaning that if your mount left the area of the dome but you did not, the dome would still end.
And of course, this is all predicated on your ability to use a Medium creature as a mount because your mount must be:
A willing creature that is at least one size larger than you and that has an appropriate anatomy can serve as a mount
So you must be a Small creature or else the only mounts you could summon would be Large and therefore break the hut.