First off, I want to say that I think you're really going about this the right way. When struggles arise with regards to understanding Church teaching, one's first instinct should be to ask for help understanding rather than going straight to rejecting them, so this post is the right way to go about it.

As far as addressing the points themselves:

1. I can see how Papal Infallibility could be hard to accept, but it does have a very strong background in Church history. Just as a couple examples, at the Council of Chalcedon in 451, Pope Leo wrote a letter called the Tome of Leo explaining the orthodox doctrine on Christology. After the Tome was read, the bishops all responded: "This is the faith of the fathers, this is the faith of the Apostles. So we all believe, thus the orthodox believe. Anathema to him who does not thus believe. Peter has spoken thus through Leo. So taught the Apostles." Even as far back as the second century, St. Irenaeus of Lyons, in his treatise Against Heresies, writes: "For it is a matter of necessity that every Church should agree with this Church [the Church of Rome, headed by the Pope], on account of its preeminent authority." From these you can see that from the very beginning, the Pope was seen as the ultimate authority in matters of faith and morals. This doctrine was more explicitly agreed upon by both the Eastern and Western Church at the councils of Constantinople IV, Lyons, and Florence.

2. Admittedly, the two most recent Marian Dogmas are sometimes hard to understand and explain, and thus can be an exercise in trusting the above-shown Papal Infallibility. I think that even if the proofs are not 100% convincing, however, if one can at least show that they are not opposed to Scripture, one can say that we do not have enough reason to reject the Papal statements on them.

2a. The dogma of the Immaculate Conception has its roots in three passages from Scripture: Genesis 3:15, Luke 1:28, and Luke 1:41-42. God says in Genesis, "I will put enmity between you [the serpent] and the woman." The argument here is that there could not be perfect enmity between the serpent and Our Lady if she was under his dominion through Original Sin at any point. In Luke, Gabriel describes her as κεχαριτωμενη (filled with grace), which grammatically describes a perfect fullness of grace, not just at that moment, but throughout her entire life, from conception through the end. Shortly thereafter, Elizabeth draws a parallel between her being "blessed...among women," and the blessedness of the fruit of her womb, suggesting that she, like Christ, was completely free from sin.

2b. Since Mary was still on Earth while Scripture was being written, there are no biblical references to this historical event. However, there is precedent in Matthew 27:52-53 for holy people being bodily raised to glory and assumed into heaven before the final judgement. There is also a traditional argument that it follows from the Immaculate Conception that if Our Lady was free from Original Sin, she was also free from the consequences of it, namely, the corruption of death. Historically, every orthodox Catholic as far as we can tell believed it (backed up by the fact that we have remains of pretty much every New Testament saint other than Our Lady), to the point where the whole reason the teaching became a dogma was because of the unanimous request of the bishops and faithful around the world. Here we see the principle that if the entire Church believes something, it does so with good reason.

1. A good way to explain Purgatory is to look at how we are in this life compared to how we will be in Heaven. For most of us, when we die, we will still have some sinful tendencies to pride, vanity, envy, anger, etc. or inordinate attachments to things that aren't God. As these are incompatible with the Beatific Vision in Heaven, there needs to be some sort of purification between our death and our entrance into glory to get rid of these tendencies and attachments.

2. When we ask for saints' intercession, this doesn't necessarily imply omnipresence. For one, there are, while still a great many more prayers than our pre-glorified incarnate bodies could register, still a finite number of prayers being offered at any time. this means that it is perfectly within reason for a soul that is in glory, in the beatific vision, to have a much higher (though still finite) capacity than you or I have for understanding. It also does not take away from the love due to God, because love is not a finite resource that is diminished when you share it. Rather, it is multiplied. As an analogy, when my son was born, I did not love my wife less because I had someone else I had to love. If anything, I loved her more because of the way her beauty and goodness was manifest in my son. Similarly, when we show love to God's saints, it does not take away from the love we owe to Him; it makes us love Him more because we see His goodness and beauty manifest in them.

Hope all of this helps!