I'm going to venture out on a limb here and try my best at these three sentences.
The nuances here are subtle and are very hard to convey in translation. I'm going to start with the 2nd sentence you posted because it seems the easiest to explain.
I might translate this as, "have you asked them to find your Christmas Tree?" I think that's fairly literal. A bit less literal, I might put it as, "have you asked them to pick out your Christmas Tree for you?"
But this could easily be the translation for
So, what's the difference? In this case, I would say the difference is that 頼んだの is just saying "did you ask?" But, 頼んでおいたの has more of the sense of "have you put in your request [yet]?" Christmas trees sell out, getting a Christmas tree is not something you necessarily want to put off to the last minute. So, you plan ahead and make your request ahead of time, hence the use of the ておく construction.
OK. So, let's try to tackle the 3rd sentence.
I'll think about it.
The speaker could have easily said,
But that sounds like they're not necessarily taking the matter very seriously. 考えておく sounds like they're definitely going to put some thought into.
OK. So, now let's look at the 1st sentence.
When you're running a fever, it's best not to take a bath.
I suspect (and here I'd love feedback from the native speakers) that, if you said something like,
it sounds like you're saying, "When you're running a fever, it's best to quit your bath." I suspect this rewording sounds like you get in the bath and realize your running a fever and so you decide it's best get out of the bath.
And actually, I suspect this sentence probably doesn't sound very fluent in Japanese.
I think the best way to capture the nuance of the original version of this sentence in translation is to say something like
When you're running a fever, it's best to avoid taking baths.
Here I think "avoid taking baths" best captures the sense to ておく because you're thinking ahead and doing (by not doing) something, which in this case is not taking a bath.
It seems to me that the speaker could just as easily said
This is something I might say as a non-native speaker, but (again I suspect that) this comes across a bit less natural to the Japanese ear because this sentence just fails to capture the idea that stewing yourself in a hot bath is probably not what you should be doing (is what you should be avoiding) while running a fever.
This is one of those sentences that, when I hear it, I'm reminded of the subtle nuances Japanese is able to convey that are completely lacking in English. And when I hear something like this, I recognize that it sounds so typically Japanese while knowing I'd never think to word it this way. (Hence I'll never speak particularly fluently.)