Warning: wall of text! I tried to answer some of the questions you had.

Flying is a huge energy cost to organisms, so it's not always ideal to have the ability to fly. It happened to evolve in organisms that could spare the energy cost, and not in others. This is kind of an oversimplified explanation, and it's hard to explain without making it sound like evolution is a conscious process. It's not that certain species push to evolve certain features, they happen due partly to chance (as all mutations are by chance) and partly due to what happens to benefit them in the environment they live in. Penguins live primarily in water, so mutations that helped them live in water persisted rather than mutations that helped them live on land.

It's also important to keep in mind that mutations do not magically change you from one species to another. There are lots of "transition" fossils that have been discovered that show characteristics of multiple species, and they're important because they show the (very, very slow and gradual) process of new species forming through repeated mutations over time.

Another thing to keep in mind is the idea of an environmental "niche", or space. Basically, if there's room in the environment, or the food chain, for a species to grow and flourish, then it usually will. Lions are an apex predator, but the reason there are still prey animals is because that's an ecological niche that can be occupied. Plants are a viable source of food, so as long as there are plants available to eat, something is going to be eating those plants. This goes for any possible source of food - plants, animals, dung, decomposing material, bacteria, the minerals dissolved into the ocean at hydrothermal vents - if it can be utilized, some species will fill that "spot" in the food chain and use it. This concept also applies to habitat in the ecosystem itself. The reason everything doesn't just evolve the ability to fly is that there are plenty of habitats available on the ground, underground, and in water that don't require flying. As I mentioned before, flying comes at a big energy cost to the organism, so there's not much benefit to evolve flying in a species like humans that live on the ground.

Energy cost also affect how many offspring an organism can produce. In general, there's two categories: the animals that give birth to numerous offspring, and the animals that give birth to few or one at a time. Generally, the animals that produce lots of offspring at once are small, somewhat short gestation time, and there's less of a chance for each offspring to survive (example: hamsters). The parent has less energy (and parental time) to devote to the production and raising of each offspring, so the strategy seems to be devoting your energy to making as many as possible, so some will survive. On the other hand, animals that produce one or a few offspring at a time (elephants, humans, etc) are usually larger, have a longer gestation time, and devote a lot of energy and parental care into the offspring. The strategy is to devote your energy into the few offspring you do produce, to raise their chance of survival. In general, you don't see large animals giving birth to large litters because energy can't be used to produce lots of large offspring with a high chance of survival for all of them. (sorry, this is kind of a simplified explanation and not written very scientifically :( )

Anyway, my overall point is that there is a lot of science behind why animals evolve the way they do. It's not that evolution is striving for anything in particular, it just happens. The mutations that benefit organisms in the environment they live in, with the resources that are available, are the ones that persist. This means that animals won't all evolve to apex predators because that's not always the path that benefits them. There's a lot more proof behind the theory than "It happened a long time ago and stuff happens over time alot.", as you said. There's a lot of evidence from multiple branches of science, including population ecology, the geologic and fossil record, genetics, etc. that paint us a picture of how evolution happens. It's super interesting and if you want, I'm sure we can find some resources that explain all this far, far better than I could! :)

(as a final note, if anyone more qualified than I finds an error in what I wrote, let me know! I'm a student and this is all stuff I learned from various classes, but my memory isn't perfect.)