What we have here is a case of large numbers vs other large numbers, preventing a clear-cut case for high or low probability.
A simple version of the Drake equation applies here. The probability that we will be contacted by aliens is:
A. the probability of intelligent life evolving on at least one other planet, multiplied by
B. the probability that a given intelligent species will develop interstellar exploration, or intergalactic exploration as the case may be, before it becomes extinct, multiplied by
C. the probability that a given species with interstellar/galactic exploration will actually find us before we become extinct ourselves.
(I say "exploration" rather than "travel", because the aliens might not have to be in our vicinity to find and contact us.)
If you remove the "other" from A to make, "the probability of intelligent life evolving on at least one planet," its probability becomes 1 (certain), as our existence proves it has already happened. If it happened here, nothing prevents it happening elsewhere, so A is indeed very close to 1 and very likely indeed. In other words, they're probably out there.
B is where the trouble starts. Interstellar or intergalactic travel or communication within reasonable timeframes (say, between any two stars in a galaxy within a lifetime) might actually be impossible, if the universe's inherent speed limit of 300,000km a second cannot be circumvented. B could be zero, and therefore alien contact might simply be impossible.
If instead there is a way to cross the cosmos which we haven't discovered yet, we won't know how long that takes until it happens. The danger is that it requires a species to spend a very long time working from a baseline of technologies with which it might inadvertently destroy itself. We're at that stage right now; our theories of deep space travel stem from some of the same research and the same minds as our atomic weaponry. The terrible risk of this particular period in a civilisation's existence appears to lower B considerably.
Finally, C is a function of A and the possibly infinite size of the universe. If every intelligent species which will ever arise has an volume of space to itself so big it will take the rest of its lifespan to explore (likely so far, since we've found nothing in the places we can see properly), no two may ever cross paths. Since the universe is expanding, the chances of contact are shrinking all the time, especially if the species are in separate galaxies, clusters or superclusters.
Tragically, the most likely case appears to be that there's intelligent life all over the universe, forcibly segregated by the tyranny of time and distance. Of course, ET could show up tomorrow and waggle a glowing finger at our flawed view of the universe.
I think the universe is way too big for Earth to be the only planet with life, but the universe is so big that other life might never find us, and vice versa.
I don't think aliens left their mark on ancient civilisations. We recognise the pictures you refer to as spaceships or whatever because we see a lot of spaceships in fiction and real life. The original artists were probably trying for something very different.
I don't think any of the more recent evidence of UFOs is terribly conclusive proof of aliens, but a lot of them are certainly unidentified flying objects as far as the people with the cameras are concerned.