I'm not Muslim, but I know that the teachings of Jesus are a big part of Islam. Like in Christianity, in the end times it is Jesus who returns to take the faithful to heaven, according to the Quran. Like in Judaism, the Quran does not teach that Jesus was the son of God (the Quran states he was a prophet). Islam is one of the three Abrahamic religions along with Judaism and Christianity. The similarities in the three religions outweigh the differences.
Obviously Muslims do not believe that Jesus is the Lord and Savior and path to Eternal Life, as outlined in the Christian New Testament. However, he is a prophet within the Islamic faith, and many of his parables and teachings are preached within the religion. Teaching people to not judge, to love, and to serve one another are concepts no religion has a monopoly on.
So it would make sense that we would follow most of his teachings. The only major contradictions that we muslims have is that Jesus is the son of God. In Islam belivers of other monotheistic faith are still considered good people in the eyes of God as long as they are honest in their prayers and do not actively fight against Islam. So I guess, yeah most of the teachings of Islam and Christianiaty are very similar.
Muslims are believing that Jesus is their prophet, they called him as Isa. And the reason of Quran mentioned Jesus more times than Muhammad because Qur'an mentioned us about everything that happens in the past for our fundamental as human. Quran also mentioned about the other prophet, not just Isa. And without any doubt, we all follow Muhammad and Quran.
Just because a certain character is mentioned more times than another in any sort of literary work does not mean such a person is more important than another. In the case of Islam, followers of the religion listen to Muhammad's teachings even though Jesus is a respected element in the stories. The number of times a person is mentioned is irrelevant.
No, Muslims do not follow the teachings of Jesus. There are some extreme parallels between the Bible and the Quran and the teachings are rather similar when laid out side by side. However, because Muslims follow a different holy man than the Christian Jesus, we cannot assume that, despite the similar teachings, they also follow Jesus.
Muslims do not necessarily follow the teachings of Jesus, and although they do see him to as a prophet. Muhammad is by far the most important and valued of these prophets. Muslims also do not believe Jesus was crucified, they believe that Allah took him up and someone else was crucified who was thought to be him.