Short answer: The rules don't say. Choose whatever you consider coolest.
Somewhat longer answer:
I would expect yes. Effects are said to travel with their subjects, so they can explicitly exceed for example their range once they've been cast. An effect with a non-concentration duration is therefore a feature of the subject and should share its frame of reference. You obviously can't actually issue commands to the subject while it's in stasis.
That's my expectation, but I think the real answer depends on how you conceptualize temporal stasis. Ask yourself this: Can a subject of temporal stasis be stripped of its gear? Neither the spell nor the rules governing targeting specify, but I consider the answer to be "no". And I construe an ongoing spell effect as just another thing that is temporarily part of the subject, but that's not the only valid interpretation of events.
Spell durations are unaffected by a sequester cast on the same target.
Spell Duration (PHB p.203):
A spell's duration is the length of time the spell persists.
This has no stated dependency on the state of the target, including how time is flowing for it. It is an independent thing.
The sequester spell states (PHB p.274):
it [the creature] falls into a state of suspended animation. Time ceases to flow for it, and it doesn’t grow older
Therefore time ceases to flow only for the targeted creature itself, not the environment it is in and, by implication, not for effects that are on it.
There is also the following clause:
This spell also ends if the target takes any damage.
Which means a sequestered creature can take damage despite the ceasing of the flow of time, though this ends the spell which allows the damage to have an effect. Apart from that, however, time has ceased for the target and so you cannot change its state. Healing will not work, you can't put it in a silly pose. As the time ceases effect only applies to creatures not to objects, it would be a DM's call if you can strip a creature of all its belongings, in the same way certain spells don't work against objects that are carried or held.
Spell durations will run down, they will try to effect the creature and fail (e.g. healing will not work, telepathy will not work etc.) until the sequester ends.
It is clear that the "time ceases" part of the effect is clearly intended to allow a creature be sequestered for any length of time without dying of old age, but it is stated that "time ceases to flow" until the sequester ends due to a clause, a dispel or the target taking damage.
Don't bother sequestering a creature that is currently suffering from a spell or effect that has an overlapping duration and damages it during that duration, the sequester will end the next time the damage is applied.
As an aside sequestered objects (and plants) are effected by time passing, as it is only creatures that the "time ceases" effect applies. Iron will rust, food will rot, plants will grow.
Best Answer
The creature can't perceive its surroundings or take actions
The character is stuck in time, and taking an action requires a certain timespan. If one second lasts forever, then you can't feasibly take an action.
Likewise, it can't see or hear anything, as those too would require a passage of time. As far as they are concerned, the moment they go to sleep, and the moment they wake up, are one and the same. No time has passed in their mind.
However, that does not mean the creature is incapacitated, as nothing in the spell says that the creature is incapacitated.
The reasoning "it can't take actions or reactions, therefore it's incapacitated" is flawed in itself. "Bob has disadvantage, so he must be blind!" could be true, but it doesn't have to be, he could be poisoned, cursed, fatigued, etc.