Unlikely, yes. Impossible, no. It's like the premise of putting a bunch of monkeys in a room with typewriters, on a long enough timeline one of them will eventually produce a copy of a Shakespeare play just by sheer chance.

We don't know what came before the Big Bang, or if there was anything before it. It's also possible that there was no "before" because space-time as we know it may not have existed prior to the Big Bang.

To me, it's just a difference in mindset. Science recognizes the limits of what we know, tries to quantify what we know we don't know, and seeks to learn and understand the things we don't yet know. Religion often uses "God" as a catch-all to explain everything we don't know. To paraphrase NDT, this makes God an ever-receding pocket of ignorance. I'm ok with that.

What I'm not ok with is dismissing scientific findings in favor of accepting a storybook explanation on faith alone. One of the key tenets of the scientific process is that your hypothesis must be disprovable. The requirement to accept things on faith is antithetical to that idea, you can never disprove the existence of God. That alone makes the case for God's existence extremely weak to me. We can however disprove the creation story in Genesis if we're examining it literally. It's easily provable that the universe existed for more than a handful of days prior to humans. It's provable that life on earth existed for millions of years prior to humans. It's provable that light existed in the universe for billions of years prior to the existence of earth, life, and humans.

I don't take much issue with people holding their own personal religious beliefs, but I don't think it should override what we can observe about the world by hard evidence. I think religious texts that reference creation or history better serve as metaphors or parables than literal truth. It does also make me uncomfortable that children raised in religion are often taught to accept these stories as literal fact on faith alone- I think it hinders critical thinking and curiosity, and leads to a populace that is gullible and prone to accepting false narratives in ways that are more directly damaging to society.

I may have digressed a bit there, but to directly answer your question, I "believe" in the Big Bang to the extent that it is the best explanation of the earliest point in the existence of our universe based on our current knowledge. I am open to, and even expect within my lifetime, that the theory of the Big Bang may be modified as we gain further knowledge, or even thrown out completely if we are presented with contrary information. That's how we examine the world scientifically, my "belief" in the Big Bang is not unshakeable and I don't see changing my views based on new information as some kind of assault on my beliefs or admission of defeat. Rather, any new information that refines or changes our understanding is a victory. If God descends from the heavens tomorrow and publicly states that he created the universe exactly as is detailed in Genesis, and that all of the contrary evidence was falsely put in place as some kind of test of faith, I will readily accept it. I just feel the need to clarify "belief" in things I can observe, measure, touch, etc from "belief" in the context of taking something entirely on faith with no testable evidence. Have you ever heard of the idea of "Last Thursdayism?" It's sort of a thought experiment, that the entire universe and everything within it came into existence last Thursday, and everything you know of the time before that is just false memory you were created with. It seems silly on its face, but there's no way for you to disprove it. How is that any less valid an origin story than Genesis? Because it wasn't written on some old scrolls in the desert? How do you even know the scrolls or the desert existed before last Thursday? You could test the age of the scrolls with radiocarbon dating, but that method can also be used to disprove the Genesis story. You could present geological evidence that the desert is more than a week old, but I can present geological evidence that the earth is older than 6,000 or 12,000 or however many years old the Bible says the earth is this week. I'm not trying to mock your beliefs, just attempting to demonstrate that fairly comparing the theory of the Big Bang to a religious creation story is impossible, it's apples and oranges. You can compare the creation story of one religion to another, or even to any made up idea like Last Thursdayism, but even then you'll find it difficult to make realistic conclusions about which is more valid. Are we basing it on who wrote it? How long ago it was written? How many people believe it? None of these measures really indicate that one is more or less valid than the next.

Based on the fact that we have fossil records and now understand genetics to the point that we can prove the path of evolution that led to humans and that all creatures on earth share a single-celled ancestor, I don't think it's reasonable to think that humans originated in something close to our current form off-planet. However, the idea of panspermia is interesting, that life on earth could've been seeded by building blocks that arrived via meteorite (or were intentionally placed by an intelligent alien species.) One of the questions that comes up when speculating on how common life is in the universe is that, as far as we can tell, life only arose on earth once. All life that exists and has ever existed, that we know of, share one common ancestor. If life were common where conditions were suitable, it seems we would've likely found evidence for multiple instances of genesis of life here on earth. Perhaps our planet was intentionally seeded with the right building blocks for life to form, or even with a cocktail of ingredients tailored to conditions on earth.

There are other smaller critiques that can be made toward your thought experiment, such as any craft that could crash into the earth with enough force to cause the mass extinction of the dinosaurs, yet remain structurally sound enough to allow for even a few of its passengers to survive, likely would be fairly easy to detect. Further, we have found evidence in the geological record that supports a meteorite coinciding with that mass extinction, in the form of a worldwide deposit of iridium from the meteorite as it burned on atmospheric entry (iridium is either extremely rare or non-existent naturally on earth.)

The stories in various cultures of gods with seemingly magic powers, or flying machines, or many other things that can be explained by technology rather than magic today is interesting, and I don't think it's entirely impossible that some could have been ancient aliens or members of some civilization that was technologically advanced far beyond our comprehension at the time, but popular culture and the "crank" nature of some of these ideas' most vocal proponents has understandably put most people with a professional reputation to protect off of these ideas. It's interesting to ponder, but I doubt we'll ever see any solid proof of any of it.

Some movies have touched on these ideas, Prometheus opened with a scene of an alien forerunner intentionally seeding a barren, early earth with the precursors to life. Mission to Mars also may be of interest to you, but I won't go into detail to avoid late-story spoilers. One of the Predator movies also showed a flashback to an earlier time when they visited earth and were received as gods. A relatively small part of the movie, but still cool to see it played out on screen. I'm sure there are many more sci-fi stories that explore these ideas, but those are a few relatively recent movies that come to mind that you may enjoy, even just for curiosity's sake.