I'm a native speaker, and we use "get ready" in exactly the way you describe.
Even though we sometimes qualify the phrase by saying: Get ready for [some event], that "for" clause can be omitted when the listener already knows where we are going. In other words, although get ready can mean different things in different contexts, I can still usually say:
C'mon! We're leaving! You need to get ready!
and that will be understood to mean: put your shoes on, brush your teeth, comb your hair and look presentable, etc.
The precise meaning get ready will vary, depending on where we are going. So long as the hearer knows where we are going, I don't need to add the, say, for your piano practice. For example, I have teenagers at home, so, for me, it works very much like this:
10 or 15 years ago, though, whenever I said, "Let's get ready" to my wife, that always included "Make sure we have the diaper bag!"
You needn't worry about get ready being misconstrued to mean "brace yourself for some bad news" or "get into your starting blocks." Those are very specialized meanings that people wouldn't even think about, unless they were in a situation where that would be the meaning that makes the most sense.
For example, if you wanted your picture taken, and I was holding your camera, I might say, "Get ready..." – which doesn't mean, "go put on your shoes," or, "brace yourself for some bad news." It simply means, "Get ready, I'm going to snap the picture now" – so you should probably just give me your best smile.
What would happen if I used "not yet" in these sentences...
Both "not yet" and "still not" are fine, but they carry different nuances. "Not yet" implies, strongly, the notion of "but that we clearly expect to happen somewhere in the future." "Still not" is very much weaker in that respect. So each has the following sense:
1') I have not yet found what I'm looking for, but I'm determined I will.
versus
1) I still haven't found what I'm looking for, and I'm just about ready to give up trying.
Curiously, that difference is much reduced in your examples 3 and 4. Neither of those uses of "not yet" carries the same kind of inevitability that the phrase does in the earlier examples. In fact you could probably interchange "not yet" and "still not" with little or no change in meaning for points 3 and 4.
Best Answer
The idiomatic expression is "race against time".
To race with something or somebody can also mean you are competing against them, so arguably it does mean the same thing. But idioms are recognisable and when they are changed it is noticeable and quite jarring.
This ngram of the two phrases shows that the idiomatic expression is well used and your alternative barely registers. So the direct answer to your question is no, it isn't the idiomatic choice.