So if the external factors remain subject to dukkha, and we are stuck with them, how does the cessation of dukkha actually happen?
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I think this was discussed at length. 
Sorry I can’t find the posts now.
Hi Martin I would contemplate first why is dukkha (in relation to existence) perceived as dukkha first. If you think (for ex: old age) dukkha is not dukkha then going from there to cause of dukkha, cessation and path will be complicated. Also we have to remember that dukkha arise within us. Nothing external is labelled as dukkha. I am not saying you have no idea about it. But if in doubt go back to step one and try to clear the remaining doubt in there to 0% first. I hope the suttas and watching your thoughts will help. 
Agree.
That is why we have to eliminate craving so we will not stuck with external or internal.
Good post.
My assumption has been that dukkha is purely “internal”, and that dukkha fully ceases for the Arahant.
But have another look at the OP sutta, which says that sense-objects (the “exterior”) are also dukkha. This raises a question: How does dukkha cease if sense-objects are inherently unsatisfactory, due to anicca? How does dukkha cease while sense-objects continue?
If you check the adjoining suttas, you’ll see that dukkha here is one of the three marks of existence.
I see your point.
For Arahants only the Dukkha Dukkha applies. In my opinion, Not other two.
As for whether Arahants still experience dukkha, note the refrain in AN10.28: “In this very life he makes an end of suffering”.
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“My assumption has been that dukkha is purely “internal”, and that dukkha fully ceases for the Arahant.”
Since the cause of dukkha is craving, when craving stops dukkha stops. For arahants it’s the craving for existence that has stopped. So (according to my understanding) exterior sense bases can still generate feelings in an arahant but it doesn’t give rise to craving and clinging to create new existence moments that will fuel the samsaric journey.
For example if an arahant’s dukkha is pain because of old age or sickness you can’t really say an arahant doesn’t feel it because he has let go of craving (i.e dukkha has ceased). In my knowledge then he is a dead man. 
IMO he still feels without a reaction with a self view.
“OP sutta, which says that sense-objects (the “exterior”) are also dukkha. This raises a question: How does dukkha cease if sense-objects are inherently unsatisfactory, due to anicca?”
The way I read the sutta,
“Sights, sound, smell, taste,touch,thoughts (external sense bases) are suffering.”
(Who sees, hears, etc.,? Five grasping aggregates. Why is it suffering? It can’t control those the way it wants)
“The cause and condition that gives rise to sights also suffering.”
what gives rise to sight, sound, smell etc.,? 5 grasping aggregates.
Since sights are produced by what is suffering,(grasping aggregates) how could there be happiness?
[4.1Now this is the noble truth of suffering.4.2Rebirth is suffering; old age is suffering; illness is suffering; death is suffering; association with the disliked is suffering; separation from the liked is suffering; not getting what you wish for is suffering. In brief, the five grasping aggregates are suffering. ]
SN 56.11
Perhaps my wording is not aligned with the suttas. So ignore if it is confusing.
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Agree.
Arahants do experience pain but not Dukkha.
I see this in terms of the Arrow Sutta, ie the Arahant is subject to bodily pain (first arrow) but immune from mental anguish (second arrow).
But I’m not sure this addresses the problem raised in your OP, ie dukkha as one of the three marks, dukkha as an inherent quality of exterior sense-objects.
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A few possibly useful suttas
“There are, O monks, these three feelings: pleasant, painful and neither-painful-nor-pleasant. Pleasant feelings should be known as painful, painful feelings should be known as a thorn, and neither-painful-nor-pleasant feelings should be known as impermanent. If a monk has known the feelings in such a way, it is said of him that he has the right outlook (sammaddaso). He has cut off craving, severed the fetters (to existence) and, through the full penetration of conceit, he has made an end of suffering.”
Tisso imā bhikkhave vedanā. Katamā tisso: sukhā vedanā dukkhā vedanā adukkhamasukhā vedanā. Sukhā bhikkhave vedanā dukkhato daṭṭhabbā, dukkhā vedanā sallato daṭṭhabbā, adukkhamasukhā vedanā aniccato daṭṭhabbā. Datthabba Sutta: To Be Known
"‘Feeling should be known. The cause by which feeling comes into play… The diversity in feeling… The result of feeling… The cessation of feeling… The path of practice for the cessation of feeling should be known.’ Thus it has been said. In reference to what was it said? Nibbedhika Sutta: Penetrative
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