Relationship between the 5 higher fetters and the Three Characteristics or the Three Doors I use the definition of Three Doors posted here: https://www.reddit.com/r/Buddhism/comments/6rrbl8/any_commentaries_on_the_three_doors_of_liberation/dl7h0e1/

To me, it looks like there is a connection:


Desire for existence in the realm of form
Desire for existence in the immaterial realms


They are about the tension between being and non-being. In other words, existence or non existence of the self, and that pretty much is Emptiness of the self.


Conceit


The conceit "I am... (this or that)". This looks related to Signlessness (there are no inherent attributes). 


Restlessness 


Seems related to Wishlessness.


Ignorance


This kind of encompasses everything.

Of course, the Three Doors are related to the Three Characteristics too.


Emptiness -> no-self.
Signlessness -> impermanence.
Wishlessness -> unsatisfactoriness.


I am understanding the fetters wrong and making up the relation or does this makes sense? I'm trying to understand them as good as I can.
Like they say on reddit, the 3 doors are not in the suttas. The 3 doors are some inventions from the commentators of some mix of commentaries, abidhammas, and suttas. 
Those are not the dhamma. Even the mix of commentaries, abidhammas, and suttas on which they comment is not the dhamma. Even the suttas alone can be  blindly said to be the dhamma.

So first, the usual way use of doors is the ''guarding of the sense doors''.
For the signless, there is plenty of that in the suttas, but there is not much about them, just like for ''desire for non-existence''. They are just listed and zero explanation. This is great for commentators who can tack their various speculations in their teachings, but at the end of the day, nobody knows what those means.

So for the signless like here https://www.accesstoinsight.org/tipitaka/kn/snp/snp.2.11.irel.html

the note says


  The Signless (animitta) is one of the three Deliverances (vimokkha) by which beings are liberated from the world. The other two
  are Desirelessness (appanihita) and Emptiness (sunnata). The Signless
  is connected with the idea of impermanence of all conditioned things
  (cf. Visuddhi Magga, XXI 67f).


This is the usual interpretation of those weird signless, desirelessness and emptiness liberations. You can read the Visuddhi Magga to read more on this, but the Visuddhi Magga is not the dhamma, only a some commentary by some people.

The usual connection with the fetters is with the births, ie Breaking the first 5 fetters frees you from rebirth in the sense-world, and breaking the remaining 5 higher fetters, frees you from being reborn in the form world and the formless world. 
So the non-returners have broken the 5 lower fetters and if they fail to reach nibanna as humans, they get to be reborn as some non-returner devas. Now the difference between those nonfully enlightened devas and the normal devas is here https://suttacentral.net/an4.123/en/sujato 
https://suttacentral.net/an4.124/en/sujato

that the normal devas delight in whatever world they live, then they are reborn in hell sooner or later (like any non enlightened people), and the enlightened devas just die like the human arahants.

So to link anicca dukkha and anatta to the fetters, you link them to the realms of births and you can say that the five higher fetters are failure to see anicca, dukkha, anatta  of those higher realms , plus the failure to see that what is anicca about those realm is also dukkha, and to fail to see that what is dukkha about those realms is also anatta [ie to to crave for them].