If the slave doesn’t die within 3 days but let's say after 5 days, will the master be held accountable?
No, in Moses' law there appears to be zero accountability (for the charge of murder) unless the slave dies of his wounds within two days of the beating. Note that this statute addresses a specific behavior: striking one's slave or bondservant "with a rod", a wooden stick.
Does the slave have the same rights as a non-slave?
Absolutely not! Slavery statutes in the law of Moses seem repulsive and barbaric in light of New Testament grace and truth and the new egalitarianism Paul preached wherein "there is neither bond nor free" (Gal 3:28), but what do you expect from a body of civil, criminal, political, military and religious code enforced by capital punishment that essentially made nothing perfect, and which was never designed for a righteous person to begin with? To the chagrin of many apologists, austere slavery is enshrined in Moses' law.
Will the slave be still his property as the text indicates?
Yes, there being no stipulation to the contrary that I can see pertaining to the question of beating with a rod as a form of discipline and punishment. Beating certain subordinates with a rod was an idea well ingrained in the Jewish psyche at least through Solomon's day, as witnessed by Proverbs 10:13 ("a rod is for the back of him that is void of understanding"), 20:30 ("the blueness of a wound cleanseth away evil: so do stripes the inward parts of the belly"), and 26:3 ("a whip for the horse, a bridle for the ass, and a rod for the fool’s back").
That [Exodus 21:26-27] indicates that the slave who has been injured and didn’t die is no more the property of the master?
If you're asking this question in a general or conceptual sense, the answer is no. Injuring your slave in the course of beating him only put you at risk of losing ownership in two specific instances: loss of an eye, or loss of a tooth. Otherwise he remained your property, and if he suffered permanent injury in some other part of his body, you would have to absorb the related expenses and losses, hence verse 21 says "if he continue a day or two, he shall not be punished: for he is his money."
[this] seems strange because verse 12 says, "Anyone who strikes a person with a fatal blow is to be put to death..."
To my understanding, the law differentiates between the kind of beatings that ought to be administered (in which mercy rejoices against judgment, as in "if thou beatest him with the rod, he shall not die," Prov 23:13), the kind of beatings that are lacking in mercy and restraint (resulting in loss of life, tooth, or eye), and the kind of beating that would fit the category of murder.
I should add that in my view Exodus 21:12 and 21:20 are talking about the very same thing: capital punishment for murder, however verse 20 specifically addresses the murder of a slave, making it clear to all that a slave owner will face the same punishment as any other murderer (unless, as verse 21 implies, the slave happens to die sometime after the two-day statute of limitations expires, in which case the owner had no liability).
21:12 He that smiteth a man, so that he die, shall be surely put to death.
21:20-21 And if a man smite his servant, or his maid, with a rod, and he die under his hand; he shall be surely punished [i.e. put to death]. Notwithstanding, if he continue a day or two, he shall not be punished: for he is his money.
All references KJV