There is no push to save others, because no one is viewed as lost. There is no concept of original sin or people being born in need of saving. Judaism isn't alone in this, there are many groups, often among ethnoreligions, that do not encourage or just do not take converts at all or view that there are paths for others.
In most traditions, the 'chosen people' concept is chosen to be the example to other nations through having additional responsibilities. This comes from a cultural time and place where that was pretty normal, we see remnants in the Bible of the ideas that different nations had different gods and one's behaviour and practices were an example to others and a sign of the Divine's favour in some conflicts between Israel and others. We also see remnants within Christanity of the idea that one's behaviour should be an example and proof that one is following the true path.
Within Judaism there is the concept of Noachides - gentiles who follow the laws of Noah and through them connect with the Divine - though currently it's largely only Haredi Jewish groups including Hassidics pushing this as a conversion path. There is conflict between these groups and some Orthodox groups who feel it is better to fully convert that treat Noachide as a separate group to convert to. There are far fewer Noachide laws, most put it at 7 with some sublaws compared to the full 613, but by most Rabbinical traditions, they would ideally be enforced more harshly post-Messiah. Other branches largely don't recognise Noachides as it's part of the Rabbinical tradition that some view as historically interesting, but not religiously relevant. There are also gentile prophets, most known are Job and Balaam, who are sometimes discussed as Noachides though the latter having a rather mixed traditions and interpretation, but all are traditionally meant to be a sign of connection with other nations.
Of the Jewish traditions that have an afterlife, everyone goes to the same place. Some later traditions have the concept of a period of purification where people's souls are cleased of their suffering and sins, lasting no more than a year (which is why the Kaddish prayer is said for 11 months), and in the end either rise to be with the Divine or fade away into shadow and nothingness. There is no concept of eternal torment or an eternal paradise where we are individuals as we are in life, how long the fading takes depends on the tradition with the Biblical references to talking with the dead.
There is a view that there are potential Messiahs in every generation, but that all so far have failed. There are many lists out there of failed messiahs, some include Jesus in that though around his time there were many messianic cults due to the oppression by the Romans and there are many including gentile King Cyrus who are called messiahs, but not The Messiah. Simon bar Kokhba is traditionally viewed as the person who has been the closest to meeting all criteria, though Hassidic Jews who have a passionate belief in the near end times sometimes put Rebbe Schneerson as the closest, with a dwindling group either refusing to believe his death or that he will return (there is no traditional belief in a messiah who returns from the dead, but there are still groups who have used it through time). Hassidic's end of time beliefs is viewed by some as to why they push the Noachide concept as much as they do, due to the writings on people from every nation coming to glorify the Divine at the end times - if you convert, you're no longer considered of the other nations (that is also part of why some Orthodox groups are vocally against Noachides being promoted, while they also believe in the coming of the Messiah, they view the Hasidic pushes as more harm than good and not as transparent as Orthodox group view it should be).
With the laws on sorcery mean actual sorcery, not tricks, largely on the view of where the powers come from. The Bible has people who are able to call up the spirits of the dead to talk and texts within and other archaeological evidence show a belief in evil eye and being able to call upon evil spirits/'night creatures' to attack others, particularly babies. It falls under not having other gods before Him. While modern Judaism is strictly monotheistic, ancient Judaism in practice was more monolarity (worshiping one while acknowledging the existence of others) with pockets of henotheism (worship of one while accepting others in other areas) and polytheism with a Divine Council structure (one deity, sometimes with a consort, over other powers) which was very common in West Asia at the time - most of the names of the Biblical God are shared with or have roots that come from Divine Council systems in the area that are viewed by some to have merged over time and interaction.
As an ethnoreligion, there is a difference between ethnically Jewish and following faith, but due to Christanity's very active and intentional separation from Judaism and the brutal persecution of Jewish people by Christianity and Islam over centuries, it is largely considered in many Jewish groups to be a rejection of both to take on those faiths. Some exceptions are made for groups like Unitarian Universalists or Quakers who, while from Christian traditions, do not require a belief in Jesus as divine and like all groups there are differences in opinions on the individual level.
It isn't really appropriate to take the history of early Christianity coming out of Judaism and Greco-Roman groups, and that some early Christians also viewed themselves as Jewish or that some Jewish people knew the Christian writings well enough to mock them, and apply that to modern Jewish groups as evidence that Messianic Jews should be treated by all Jewish groups as still Jewish, just as we cannot apply ancient Judaism's monolarity to modern Judaism's monotheism. The cultural shifts of Christianity out of Judaism happened and is an important part of both faith systems. It was an active part of the early church to reject Judaism, the many years of work put into separating their calendars to make themselves apart and their choice to include Paul and psuedo-Paul texts as doctrine which emphasised their separation over other texts that didn't, just as much as many Jewish groups at the time actively rejected Christianity and other groups that pushed to shift away from Rabbinical authority and traditions as the Sadducee sects (which also rejected Rabbinical authority) faded away into other groups and since maintain the separation from groups that actively and at times violently treat them as separate.