You sound a little bit like me and "Ne" is my strongest function-in-attitude. The descriptions of personality types will never be sufficient because we can see a lot of problems with them immediately and we can also easily perceive ourselves as many different types.
Relating to all cognitive functions seems to me to be an issue N types in general can suffer with. It is too easy to read about them and learn about them and perceive yourself a type of that function, or with it as auxiliary, or tertiary, or inferior. This is a problem for me as well. "Ne" in function-in-attitude tests, INTp-1Ni in Socionics with a 99% chance of ENTp NeTi as well. When intuition is too strong we get detached from reality and find it too easy to see ourselves as this or that.
The problem here I think is to work on the Thinking or Feeling function to help cut out some of the BS that intuition perceives and try to see what actually makes sense. Introduce some S to help narrow what makes sense for you down with actual data, do this by seeking out perspectives from friends and family. See how other people perceive you. I do not know how accurate this is as I am still working through typing myself and gathering more information like this to help do it. Strong N and a weaker T/F will make it hard to type yourself in my opinion.
Bad memory is not necessarily proof of inferior Si, but can be indicative.
Si is my third function but (I think) I have a very good memory. I do forget things, but they are usually the things I have decided are not as important. Faces and names rank high amongst these 'unimportant' details.
Memory has nothing to do with type. Jung said this himself. This idea that Si is memory-related is incorrect. It does make sense in a roundabout way in that Si types build up a database of subjective sensory impressions over time which means in real-time, new subjective impressions they perceive will be compared at some level to previous subjective impressions they have perceived.
This is no more memory related than how any other introverted function works on account of introversions referral to yourself and your own experience. I have to say here that introversion itself does not equate to "better memory". I am talking about how the functions work on a fundamental level and not how good anyones memory is.
Is being scatterbrained, carefree, reluctant (And therefore dropping completely) to follow chores like exercises are indication for inferior Si?
It depends on the model. Sure it fits some models interpretations of inferior Si. These models probably have no actual scientific support. If you look at this from an MBTI perspective which only has the dichotomy in use for typing people with then everything you mentioned would fall under P and only P.
In Socionics which in my opinion seems to get a lot of things right what you mentioned call fall under the INTp type for example because they correctly attribute these "P" characteristics to the introverted intuitive type. In MBTI you could not be scatterbrained, carefree and reluctant to do chores and still get a J result, you are choosing things there are are directly related to an MBTI P.
I am not saying you are an INTp in Socionics I am only highlighting how the characteristics you suggested will speak to different things in different models. In MBTI they will speak to P types and if you are an introvert this for no reason will mean you are "Ne" or "Se" because P relates to your perceiving function being extroverted. In Socionics the same characteristics would produce a "p" result and this will relate (accurately in my opinion) to your
dominant function even if introverted.
Something to keep in mind is how most models that follow in the footsteps of the MBTI twist functions very far away from what they are. Personalityjunkie for example has N so far removed from intuition that it is Thinking. Much of everything that personalityjunkie says about N is T. They also relate too much of what is introversion to N and too much of extroversion to S. So you have to be careful about the model.
It is interesting to learn about them all but trying to find a type that is consistent across all models is a fruitless endeavor and does not actually make sense because of how different each model is.