The Sabbath is the very first of God's Appointed Times. In Genesis 1:1-2:3, God sets a pattern for us to follow by creating everything in six days (working) and then resting on the Seventh. The Seventh day of the first week was the very first day that God set apart (made holy). Later, he codified it as a mandatory rest for all (Exodus 20:8-11), even rest for the animals. In fact, God commanded that even those not in covenant with him observe the Sabbath. It's that important to him! So we've basically answered both questions from the title, but some may object and say that the Sabbath was changed by either Jesus or his Disciples. So let's address that claim.
The first question we should ask is this: Does Jesus, or his Disciples have the authority to change the Sabbath? Some may say that Jesus does, because he is God in the flesh. And while it's true that Jesus is God in the flesh, he also lived as a man on the Earth. In order to live a life without sin, he must keep God's Law. In Deuteronomy 12:32, it says to keep God's commandments. And forbids us from adding to or diminishing from them. Changing the Sabbath is not keeping God's commandments, but replacing them with your own commandments.
Also, we have that Deuteronomy 13 Test to worry about. Deuteronomy 13, in a nutshell, is this: Anyone who teaches what is contrary to God's Law is, by definition, leading people into idolatry. And anyone who teaches obedience to God's Law is, by definition, teaching people to serve God. Would God tell the Israelites to test the teachers in this way, then send someone to teach the opposite 1,500 years later? Those who know what God's Law says would immediately conclude that such a person was a false teacher. In fact, this is what the Pharisees tried to do so many times in an attempt to get rid of Jesus. But they couldn't find anything. Ironically, the high priest, after hearing Jesus say that he would be at the right hand of the father, broke the Law by tearing his garment (Matthew 26:65)! By the way, did I mention that it was considered blasphemous for anyone to claim that they would be with God in heaven?
So, while you can argue that Jesus did have the authority to change the Sabbath, doing so would invalidate his work on the cross and our faith in him would be futile. So if even Jesus did not have the authority to change the Sabbath, then his Disciples certainly did not have any such authority. So if they didn't change it, who did? False convert Constantine did. Constantine is the one prophesied about in Daniel 7:24-25.
So, what about the evidence that the Disciples observed the first day of the week (what we call "Sunday") as the Sabbath? Well, in Acts 20:7, it mentions that Paul was speaking until about midnight, and that it was the first day of the week. The way that the Israelites count the starts of each day is different than from how we count them. Their days start at sundown the night before. So Paul was still speaking on what we would call "Saturday night", going into Sunday.
In 1 Corinthians 16:2, Paul instructs one of the congregations to collect money on the first day of every week, so that they will not have to collect money when he arrives. This is typically interpreted as a change in the Sabbath, but since Deuteronomy 13 calls that idolatry, the collection on the first day of the week is so that they will not be collecting on the Sabbath.
And before anyone tries to use Hebrews 7:12 as evidence that the Law was changed, Deuteronomy 12:32 and Deuteronomy 13 prevents this from happening. The passage is greatly mistranslated. The Greek word for change is "allasso", but the words translated as change are "metatithemi" and "metathesis", which both mean "transfer". It's kind of difficult to change something that is eternal.
So the Sabbath is a day of rest (not a day of worship), and it is on the seventh day of the week, which is the day that is called "Saturday".