I know that there are animals that are "simpler" than other animals but are there any that are half-evolved? Why aren't there living half ape and half humans?
Oh come on. You know if Australopithecines or Homo habilis still existed you would be asking "Why aren't there living half Homo habilis and half humans"? And when the other Great Apes go extinct you'll be wondering why there are no transitional forms between humans and monkeys. The answer to that question is, humans are apes; chimpanzees and we are pretty much as close as two species can be; we could have closer forms that survived but we could also have a much bigger gap between us and our closest relatives than we currently do. In other words, any ape is a valid example of something "half-human half-ape". It's like asking for a vehicle that's half-car, half-volvo.
Are there animals that are just starting to evolve arms and legs?
You mean, modifying fins into limbs in a general movement from water-living to land-living, like the first tetrapods are thought to have done? I like mudskippers.
Saying that every species on the planet is "transitional" because there are no ultimate or final species is an unacceptable answer because it only works on the assumption that macro-evolution is true.
Not quite, it is an explanation of what "macro-evolution" is and what things it predicts.
There are two aspects to the theory of evolution that are often confused; one is how it works to generate diversity and change in living organisms, and the other is how it works to adapt living organisms to their environment and create what we humans see as "design", "function". Both things are results of evolution, but the latter is much less important than we'd think, which leads people who explain evolution to de-emphasize it, which then leads people to be confused as to how it can happen at all.
The thing is, both "design" and "goals/purposes" are concepts that are meaningful in considering evolution, it's just that the words are dangerous because they imply an conscious intention that isn't there. You could replace "design" with "optimization"; random mutation combined with natural selection can form an optimizing process, i.e. a process that will over time generate forms that have specific properties. In the case of life, whatever properties promote the spread of the organism we're looking at.
As for "goal", that's just the physical fact of the world that some arrangements of matter have different effects from other arrangements of matter; meaning that if some arrangements of matter will lead organisms to reproduce better, the optimizing process of evolution will predictably lead to organisms having that arrangement of matter. So we can talk about, say, eyes being "designed" by evolution "for seeing"; the difference between that and a rock "being designed" "for having its shape" is that "seeing" (i.e. that organisms ancestors surviving and reproducing based on how well they interpreted light signals) had a causal influence on the organism's eye's structure.
So in that sense we can think there are "goals" in evolution; seeing, flying, swimming, breathing... and many lineages can be retrospectively seen as evolving "in the direction" of those goals. But the important thing is that those "goals" are completely dependent on the environment; as long as there's selective pressure for improved sight a lineage will evolve to have better eyes, but the second the selective pressure disappears or reverses, the lineage will evolve to maintain its current eyes or to lose them. And of course different lineages may have different selective pressures on them and "evolve in different directions". That's why there isn't a single overriding goal in evolution, just millions of lineages under different selective pressures that might change tomorrow.
That's why we can say every species is "transitional" (because they'll all be something different tomorrow, or not) and at the same time not be able to say transitional towards what (because the environment is complicated and we usually can't tell under what sustained selective pressure a given organism is under, and even if we could we couldn't be sure it remained the same over the next thousand or million years).
Having said that, when our human brains say "half-evolved" we often have specific goals and images in mind - like "halfway towards flight", "halfway towards eyes", "halfway towards limbs", "halfway towards dolphinhood". And there are plenty of "transitionals" by that definition - I brought up the mudskipper; there are also plenty of gliding animals, a plethora of eyes (or "eyes") at every point on the "good at seeing" continuum (more of a state space than a continuum really), aquatic mammals that may remind one of what the ancestors of cetaceans were like and might evolve to be more aquatic in the future...
Saying that all the transitional animals just died off also doesn't seem quite right. If all the previous transitional animals just went extinct, then wouldn't we just have a few specialized species alive today? This wouldn't allow for the diversity we see today.
Ah, but what diversity do we see today? What we observe in the fossil record is the constant extinction of some forms and diversification of others, that nevertheless remain more like each other in important ways than they were to their earlier aunts and uncles - meaning while at some point you might have forms A, B, and C, a million years later you have A1, A2, A3 and A4. To illustrate, the earliest tetrapods had a wide variety of number of digits on their limbs. Of course they all basically looked like fish. Most of those early tetrapods went extinct; the only line that survived had five digits. And millions of years later we have a stunning variety of organisms descended from that lineage, most of which look nothing like fish or each other, yet they all follow the same basic skeletal plan, down to the five digits. So which is more diverse?
Basically your question accurately describes the biosphere as the fossil record shows it, you just missed one thing: diversification. Meaning the "few specialized species" are actually a "few specialized families/orders/classes/phyla that themselves contain many species".
When thinking of evolutionary change it's better not to think of a line from A to B, but something like self-spawning fireworks. One group expands, a lot of its elements fizzling out but some of them spawning their own subgroups that expand in turn while still being recognizably part of the original group, and so on so forth with the fizzling of the most and spawning of the few, at some point the older groups might be so large and full of different subgroups that they start losing their definition, and basically at some point you realize the whole big mass of elements that now cover the sky includes both A and B, meaning if you roll back the tape you can certainly track how something that looked like A changed over time into something that looked like B, but it's really a very small part of the whole show.