Well, those aren’t particularly simple questions, Emet. The first one especially gets into a high-level rabbinic dispute in several ways!
But welcome to the forum. I’m going to move this thread to “Christology”, since your questions seem to be about that, instead of a general discussion on whether God will save all sinners from sin or not.
1.) It was not forbidden to heal on the sabbath if life was in danger. Jesus’ question assumed everyone there agreed that even an ox could be pulled from a well on the Sabbath. The point of contention was over whether non-critical healing could be done on the Sabbath, did that count as work and thus as breaking the Sabbath?
Part of Jesus’ answer was that, because the Sabbath was made by God for man and not man for the Sabbath (rabbis later disputed both sides of that), therefore God allowed and even sometimes required creatures to serve other creatures on the Sabbath. (“The priests in the Temple profane the Sabbath and yet are blameless,” was one of his examples.)
Part of his answer was that even the possibility of being a life-threatening situation allowed permission to help someone: an ox in a well wasn’t necessarily in danger of dying but might be. Don’t take the chance, help the ox.
Part of his answer was that people are more valuable to God than animals, so healing a person is that much more fitting and honorable on the sabbath. (“How much more value, then, is a man than a sheep!!?”) This was also related to the hardness of heart in his religious opponents for watching to see if he would heal a man with a withered hand while teaching in the synagogue: using the man’s illness for their own purposes in such a way!
Part of his answer was that the Sabbath is for the best of things, saving someone is an ideal thing to do, therefore it is lawful to do the best things on the sabbath. (Now Jesus says them: “I will be asking you a question: is it allowed on the sabbaths to do good, or to do evil? – to save a soul, or to destroy!?” … And looking around on them furiously, grieved at their hardness of heart, he says: “So then – it is lawful to do the best things on the Sabbath!”) Later rabbis actually agreed God blesses a person for doing the best of things on the sabbath in honor of God. There is a slightly off-color joke about this from one of the Talmud disputes, where a rabbi agreed that a man ought to please his wife (sexually) every day of the week but twice on the sabbath; thus the joke about the emphasis of doing something “every day of the week and twice on Sundays!”
So far non-Christian rabbis could dispute about this with each other, and even reach conclusions similar to Jesus. But the religious leaders knew Jesus was making high authority claims about himself, and Jesus himself framed the issue of healing on the sabbath in subtle ways to make those claims.
So, part of the answer that a rabbi might accept easily enough is that God does not rest on the sabbath but continues the action of keeping reality in existence and doing many other things, including overt miracles on the sabbath, including healing. So as long as a mere person is not doing medical work there isn’t even a question; God can be gracious on the sabbath and not profane the sabbath. But Jesus wasn’t praying to God asking to heal people (as Rabbi ben Dosa later did), Jesus was healing by direct action. This is like forgiving someone for sinning against someone else. Jesus is making a claim of power and authority on par with God Most High. (“My Father is working until now; and I myself am working!”)
Similarly, another part of his answer was that, just as God had the authority to grant His servants permission to work for food on the Sabbath (David’s soldiers even ate the bread of the presence of YHWH when being chased by Saul, though fleeing Saul would have been doing work by Pharisee purity standards in Jesus’ day), Jesus had the authority to grant his servants permission to work for food (gleaning the fields for a snack).
Similarly again, back in connection with healing on the sabbath, when Jesus commands someone he has healed to pick up his pallet and walk, he knows that’s going to set off warning flags to the Pharisees, who would regard that as possibly being work therefore forbidden in order to hedge the sabbath.
To be fair, the Pharisees didn’t always argue that this or that was certainly breaking the sabbath, only that it might be so just to be safe people shouldn’t do it: especially for the purpose of convincing God that the people were finally righteous enough for God to send the Messiah to save them! (The Pharisees used to claim that if Israel could keep the Torah perfectly for only one day, God would send the Messiah to save them from the Romans.) You should be able to see how this topic also folds back into Jesus’ claims about himself and what it means to be the King Messiah. This also goes back to Jesus talking about the ox or the sheep in the well: if you’re going to appeal to the mere possibility of something as an argument to do or not do something, appeal in favor of helping save a life and doing the best of things. A non-Christian rabbi could in theory accept that; but would not be able to accept the authority claims Jesus was making along the way without becoming a Christian rabbi.
2.) We probably shouldn’t light a fire, or do chores to strain ourselves on the sabbath, unless we’re serving other people by doing so. There are Christian Jews who keep shabbat regularly by not lighting fire on the sabbath day, preparing their meals for the day beforehand (and doing twice as much work the preceding day, by the way!) It’s a question of how best to love God and one’s neighbor on the sabbath: does lighting a fire or not lighting a fire do that best? There are different opinions either way. I for one am very grateful that other people light fires on the sabbath! – I think I should do the same to help them on their days off, in honor of God and my neighbor. (Sometimes I think God’s main purpose in having Jews, Muslims, and Christians, is so that one day we may each help provide the other with sabbath days. )
3.) I don’t know how carefully women wash themselves after a menstrual cycle, not being a woman, but it seems like a good idea. The point is to remove the blood which might otherwise rot and lead to disease and other unhealthiness: much more of a problem in ancient (sometimes even modern) Near Middle Eastern countries than here in America (in most places). If a woman is being clean, and services are being provided for her to do so, that seems to meet the spirit of the law.
4.) With God being the father, that would itself explain the contradiction: someone could legitimately argue either way that the lineage could be traced back through Joseph or through Mary, although interestingly there are legal issues not merely biological in either geneology. GosMatt traces the biological lineage from Joseph back to Abraham (with a bit of telescoping, typical of lineage tracing in that time and place, though Matthew may not have realized that since he tries to make a point out of the resulting numbers), but states clearly that Joseph was not biologically Jesus’ father. GosLuke traces the legal lineage, which usually happens to be biological but not necessarily. In that case the only salient question is why there are two different fathers of Joseph, but because of the distinction in wording the inference is that Joseph was adopted into Mary’s family as the son of Eli. (This doesn’t fit church traditions outside the scriptures, but I’m not overly concerned about reconciling those.)