Well, the problem is that the questions are formed precisely enough to be able to give a meaningful answer.
1) What is the most recent example of a newly evolved species?
That problem with the question as asked is that it presumes that there's some "instant" when a changing population "suddenly" becomes a new species. That's not how it works, so I can't tell you anything like, "on December 17th at 5:28 pm, a new species of bird appeared in Utah."
Instead, a species (or an isolated subpopulation of a species) changes slowly over time, gradually accumulating differences which eventually gain the population enough difference from either their original ancestors or their "cousins" (remaining non-isolated population) to qualify for the category of "different species".
The actual "point" where this is reached is rather a matter of taste rather than anything else (although given enough relative change there will come a time when *everyone* will agree that two different species have are definitely present rather than two "varieties" within one species). So although one population of nearly identical creatures is clearly "one species", and two populations of distinctly different creatures clearly match the concept of what people would label "two species", there's a pretty big gray area in the middle where different portions of the orginal one population are acquiring "minor" differences from each other -- and during that time there's no one sharp moment when *ping*, a distinct event occurs which suddenly qualifies the differing subpopulations as "different species starts today".
This is why there are a number of animals for which there's disagreement over whether two or more species are present, or subspecies within the same species category. Those animals are on the "cusp", in the "gray area" where they are two species in some people's views, and "not quite there" in other people's views. "Species" is a man-made category anyway, nature itself doesn't make such distinctions.
For one example (out of countless), the African Grey parrot comes in two main varieties -- the "Congo" variety and the "Timneh" variety. The Timneh is somewhat smaller, more delicately built, has noticeably darker plumage (especially the tail, which is bright red in the Congo and maroon in the Timneh), and has the Congo has a black beak while the Timneh has a "flesh" colored upper beak. There are also noticeable differences in temperament between them.
Are they two different species, or two subspecies within the same species? As one parrot-owners' website puts it: "There are two types of African grey parrots; some authorities consider them different "races", others consider them sub-species, while still others classify them as different species." Cases like this occur when differences between populations are in the "gray area" -- on the verge of having enough differences between them to eventually qualify as "clearly" different species finally.
It's kind of like baking a cake -- in the beginning it's batter, in the end it's a cake, but what is it at various times in the middle? At what "instant" does it change from batter to cake?
Similarly, plate tectonics causes mountains to rise and fall -- but there's no "instant" when a "hill" becomes a "mountain" or vice versa, and there's a "gray area" during the transition where different people will say "it's a mountain now", and others will say "still a hill, not quite a mountain yet". And it happens so slowly a "geological creationist" could say, "no one's ever seen a hill turn into a mountain or vice versa, so there", even though one could point to measured cases where hills were rising (inches per year) and mountains were shrinking (also inches per year). The creationist would then counter, "that's just 'micro-changes' -- there's no proof that the changes will go on long enough to do the 'macro-change' necessary to convert a mountain into a hill or vice versa".
Nor is it a well-formed question to ask, "What is the most recent example of a newly grown mountain?" What you *could* do is be shown mountains in different stages of the growth process, and instrument readings that show that they are indeed slowly rising even at the present time. And also geologic evidence that the tallest mountains were once much less than their current height (like sediment layers which must have been laid down flat, which are now "humped" through the mountain, or ocean-life fossils which are now up near the mountain's peak). And so on.
And so it is for for being able to make an overwhelming case for speciation, "macroevolution", common descent, and so on, and ruling out alternative explanations for one observation or another.
2) Can you name a plant or an animal of which there is only one species?
I have no idea what you mean by this. "A plant" or "an animal" is too vague. For example, if by "an animal" you mean "Moluccan cockatoo", then "there is only one species" of that type of "animal", because "Moluccan cockatoo" *is* a species. But if by "an animal" you mean "cockatoo", then no, there is not "only one species" because there are many species in the cockatoo group of the parrot family. So I don't understand the question.
If so, how long has it been known to exist without another species evolving from it?
Once you clarify the first half of your question, you'll have to clarify the second half: Do you mean "known to exist" during the past century or so, or "known to exist" across geologic time?
Still not sure what you're asking, but within the past few thousand years the domestic dog species has been evolved from their gray wolf ancestors by selective breeding (which is a directed form of, but still the same process as, natural selection). And while domestic dogs and wolves are both still canids (along with foxes, coyotes, etc.), they're no longer *wolves* -- they're a new distinct species. (Although for reasons I gave above, it would be hard to pinpoint the *exact* time that "the" transition took place even if we had a complete collection of preserved "dogs and wolves of the past", which we don't).
If this helps answer your question (whatever it was), good. If not, try asking more clearly.