During the fight with Kaguya, Obito gave his Mangekyou Sharingan to Kakashi after his death, and we know that if a Mangekyou Sharingan is given to any other person that person will achieve the Eternal Mangekyou Sharingan.
So did Kakashi achieve an Eternal Mangekyou Sharingan even if he is not a true heir of Sharingan?
In short, no. Kakashi didn't achieve an Eternal Mangekyou Sharingan.
According to wiki:
After Kakashi loses his Sharingan, his body is temporarily inhabited by Obito's spirit, giving him Mangekyō in both eyes.
So now the question is Why he didn't achieve Eternal Mangekyou Sharingan ?
Now again according to wiki:
Their vision can be restored by receiving the transplanted Mangekyō of an Uchiha with strong blood ties — ideally a sibling — thus awakening so-called Eternal Mangekyō Sharingan. The Eternal Mangekyō merge the designs of the transplanter and the transplantee's original Mangekyō.
Since Kakashi was not having his own original Mangekyo so it might be possible that he was not able to achieve it also it is mentioned that there has to be strong blood ties. So I guess according to me both of them have to be from Uchiha clan but in this case Kakashi was from Hatake clan.
I am not certain if there's any correct answer to this question.
But as for vision deterioration due to excessive usage of Mangekyou Sharingan, it was overcome due to the Hashirama cells Obito had.
Since Kakashi was having Obito's Sharingan and it has been seen in manga and anime as well, the original wielder of the Sharingan affects its nature, irrespective of its current wielder, healing Kakashi's Sharingan as well.
No, kakashi can't. In order to have an eternal magekyou sharingan, the individual must have had an eternal mangekyou sharingan implanted as the chakra shift occurs only after the user experiences the loss of one close to them at their own hands, and as obito only had a basic, not even complete sharingan, kakashi cannot
