A creature that isn't mounted is simply a creature and has all the normal mechanics and actions available to any NPC as described throughout the rules of the game. The Find Steed text doesn't assign any exceptional rules that limit what it's capable of doing or when it acts: it gets its own initiative and takes all actions, as described very well by Dale M. The Find Steed text does give pretty clear guidance on what the horse would want to do while unmounted, which is be loyal to its master and his/her telepathic instructions, again described very well by Dale M.
Regarding who controls it, there is no exceptional rule stating that the player controls it, and therefore the general rule applies which is that the DM controls all things other than the players. Therefore, the DM decides exactly what the mount does while it is 1 mile away from its master and their telepathy is cut off, and the DM also decides how the horse responds to your telepathic commands within 1 mile. Of course, if your DM plays your mount as anything but loyal to you and unusually intelligent, that's unfortunate and not intended by the RAW.
Rules As Written - Mounted
The only way a mounted mount can attack is if it's independent. If a creature is "intelligent", it acts independently. From the PHB:
While you're mounted, you have two options. You can either control the
mount or allow it to act independently. Intelligent creatures, such as
dragons, act independently.
You can control a mount only if it has been trained to accept a rider.
Domesticated horses, donkeys, and similar creatures are assumed to
have such training. The initiative of a controlled mount changes to
match yours when you mount it. It moves as you direct it, and it has
only three action options; Dash, Disengage, and Dodge. A controlled
mount can move and act even on the turn that you mount it.
An independent mount retains its place in the initiative order.
Bearing a rider puts no restrictions on the actions the mount can
take, and it moves and acts as it wishes. It might flee from combat,
rush to attack and devour a badly injured foe, or otherwise act
against your wishes.
In either case, if the mount provokes an opportunity attack while
you're on it, the attacker can target you or the mount.
People are confused about whether the Find Steed mount is intelligent and therefore independent because the Find Steed text describes the steed as unusually intelligent and assigns a minimum INT of 6. Does this qualify as intelligent enough to act independently, such as a dragon in the example? Perhaps. It isn't perfectly explicit. Confusion is very understandable.
Another confusion is whether this line from "Find Steed" explicitly overrides the mount's "independence:"
you have an instinctive bond with it that allows you to fight as a seamless unit.
Does "Seamless unit" mean that this mount is no longer independent, and therefore cannot attack in combat? Perhaps. It isn't perfectly explicit. Confusion is very understandable.
For both of these confusions, there is no basis to know for sure in the RAW, and many look to Errata and Game Designer tweets for clarification. Without external clarification, the books are genuinely ambiguous. Like all ambiguities, this leaves the decision to DM discretion.
External Clarifications - Mounted
Lead Game Designer Jeremy Crawford has issued two clarifications on Twitter.
If your DM adjudicates based on authority like Crawford, (s)he will probably want to allow the mount to attack in combat. However, with even the lead game designer changing his mind over the course of 3 years, it's easy to see what a difficult decision this is and perhaps easy to forgive your DM for choosing a different ruling than you'd expect.
My humble Assessment
As a hobbyist game designer for many years and modest statistician, the horse appears to be a valuable asset on par with a 2nd level spell slot without giving an extra attack each turn:
- Incredible movement speed, in and out of combat
- Potential to fly, depending on the mount
- Telepathic control with an animal, yielding combat and non-combat advantages (create distractions, explore areas you can't reach, have it bring you the keys to the prison cell you're trapped in, etc etc)
- An extra HP pool to scrub attacks
- Doesn't really cost a full 2nd level slot. Cast it out of combat, take a long rest, and go perhaps several long-rests before needing to recast it.
To take this and add in an extra semi-permanent attack on every single round of combat as long as the horse is alive (+6 to hit, 2D6+4 for warhorse) seems a far greater damage output than any typical 2nd level spell that consumes an entire slot on each cast and only fires once. It is my assessment that it is unbalancing and perhaps game breaking to give this. If a paladin's Find Steed warhorse lasts 1 full day and the DM gives a typical 6 encounters, each lasting even 3 rounds, the paladin gets a total 36D6 + 17 damage, for an expected value of 143 damage, though all requires hit roll. That's a conservative estimate. Compare to Scorching Ray, which also requires hit roll, and yields 6D6, for an expected value of 21. The scorching ray has the potential for more burst damage in one turn, which does not even somewhat compensate for the myriad advantages Find Steed has over it. I wonder if perhaps Crawford was on this same line of thought with his 2015 ruling. Undoubtedly it's the reason a "normal" mount can't attack at all or is too intelligent to be controlled into attacking and would probably share XP.
Finally, if a DM rules that it cannot attack while mounted, the paladin can skirt the ruling altogether and get an attack each turn from the mount anyhow by simply jumping off of it. This is clear in the rules as well as Crawford's clarifications, already shared.