Here's a sort of popular vote by internet usage: Google hits for "rightdoing" = 75,900 as of the time of this post. Notable usages include a published book title, a more modern published article, and a quoteworthy quote. Can't find it in any dictionaries myself (besides urbandictionary!), but that doesn't mean a native speaker like me would see any reason to object. (In fact, I wouldn't!)
P.S. I should add that I can't compare the hits for "right doing" because they might appear in the middle of a larger sentence like, "I don't feel right doing my comparisons like that," in which case it's not just one noun; doing becomes a separate verb.
No, they're not synonymous and they're not interchangeable.
"Nothing could be further from the truth" is a polite way of saying that something is an outrageous lie or a gross error. The meaning is literal: the statement you are referring to is not just untrue, but it is as far from the truth as you can imagine. It's something of an idiom, I guess, or at least, it's a stock phrase.
"Far and away" is an idiom used to indicate that something is not only the most in some category, but that all competitors are far behind. It is normally followed by the word "the" and a superlative. It could be something positive or something negative. Like, "Al is far and away the fastest runner in the world." That is, not only is Al a fast runner, not only is he the fastest runner in the world, but no other runner even comes close to his speed. "Betty is far and away the biggest cheater I have ever met." Not only is Betty a cheater, but no one else cheats anywhere near as much or as brazenly as she does. Etc. Such a statement may be meant literally: in my first example, Al may have just won an Olympic race leaving all his opponents far behind. Or it may be a poetic exaggeration. Al isn't really an Olympic gold medal winner, I just want to emphasize that he's really fast.
You can't substitute because "far and away" is part of an idiom. You can't rip it out and use it without a superlative. "Nothing could be far and away from the truth" doesn't make sense. Even if you take "far and away" to mean "very far", still, this would say that it is not possible for anything to be very untrue, which isn't what the speaker is trying to say at all. He's trying to say that the referenced statement IS very untrue.
Best Answer
In this context, "cherry-picking" is a very negative term. This meaning comes from statistical analysis. The term is idiomatic and informal. It is not as negative as accusing someone of lying, but it strongly implies that they do not care whether they mislead.
Suppose you are writing an article about a sports team. The team won its first game, lost its next three games, and won its last two games. Looking at the team's complete record, you could say, "The team wins about half its games."
But suppose you want to argue that the team is really good (or getting better). You could choose to look at just the last two games, and say, "The team is on a two game winning streak."
Or suppose you want to argue that the team is bad. You could choose to say, "The team lost three in a row."
Both of these are examples of "cherry-picking": There are a bunch of "facts" ("cherries") to choose from out of all of the facts ("on the tree"). Instead of doing the hard work of considering all the facts ("picking all the cherries on the tree"), you pick facts that tend to support your argument, and ignore the rest.
There are also less severe examples of cherry picking. Suppose you said "The team has won the last two-thirds of the games it played." This is still cherry-picking, but not as bad as in the previous examples.