When using "in case" as in "take this umbrella, in case it rains", you acknowledge the possibility of the action (i.e. rain). You're not certain it will happen, but it's not unlikely. You'll often see "in case" in instructions, rules, etc. - for example, "in case of fire, stop, drop, and roll".
When using "just in case", on the other hand, you emphasise the unlikeliness of the action. If you're "taking this umbrella, just in case it rains", you don't really expect it to rain, but you're taking precautions. It's also less formal, it's not something you'll see on a warning sign or in a rulebook.
"Lest" is somewhat poetic and archaic, but I think it puts a bit more emphasis on the undesirability of the consequences than the actual likelihood of the event. Don't quote me on that, though.
So you've listed a number of phrases, and you want to be able to distinguish fine shades of usage or meaning between them. I'll see if I can't help with that by using them in example sentences which are similar where possible and different where required. (I am not an architect, so don't take what follows as construction advice.) In each case, note how the rest of the sentence changes. The phrases are
- in a way that
- in such a way that
- in this way
- in doing so
- in order to
- in order that
- so that
- such that
Examples:
1: "I have designed the house in a way that allows air to circulate."
2: "I have designed the house in such a way that air is able to circulate."
3: "I have designed the house with large windows facing both east and west. In this way, I have allowed air to circulate throughout the house."
4: "In doing so" would work equally well in the previous sentence. However, they are not completely interchangeable; "in doing so" implies that an active agent has done some "doing," while "in this way" does not. "In doing so" would not work well if substituted into the following sentence:
"Crust forms in areas of sea floor spreading, and is destroyed in areas of subduction. In this way, the heat of Earth's interior is slowly allowed to escape."
For some reason, I seem to be using a thermodynamic theme for these examples. Changing gears, the next sentence would not work as well with "in this way":
"I shot the sheriff. In doing so, I signed my own death warrant."
5: "I have designed the house with large windows facing both east and west [in order to allow air to circulate throughout the house]."
6: "I have designed the house with large windows facing both east and west [in order that air may circulate throughout the house]."
The subordinate clauses enclosed by brackets in the previous two sentences may come at the beginnings of their respective sentences if separated from the rest by a comma, per the usual comma rules.
7: "I have designed the house with large windows facing both east and west so that air can circulate throughout the house."
Inverting the previous sentence, as permitted in 5) and 6), would result in awkwardness. "In order that" is more formal than "so that," from which "that" is often omitted:
"I poked a hole so air would leak out."
8: "Let Q equal the set of all p/q such that q is not zero and both p and q are in Z." [source: Wolfram MathWorld]
The last phrase in the list has a much more scholarly usage and tone than the others, owing to its origin in mathematics.
See also:
https://english.stackexchange.com/questions/126885/in-order-that-vs-in-order-to
Best Answer
First of all, for similar phrases like this, I never like to say that they are interchangeable, because that implies that they connotate the same thing. There is usually a best choice to make. While the denotation of the phrases might be interchangeable, providing an incorrect connotation could lead to confusion on the part of the reader or listener of your sentence.
So let discuss connotation with an example.
In doing so:
The first phrase in doing so gives the impression that effect of his first action (walking the dog) wasn't at all intended to cause the second. He exercised, but this occurred only as a side benefit.
In this way:
In this example, the phrase in this way is used. It is implied that Kathy desired the results described in the second sentence. She wants the hose close by. However, it isn't a very strong desire. If her husband comes and stores the hose, she will just get it out again.
So that:
In this example, Fred performed the first action (law school) expressly for the purpose of achieving the second action (get a degree). This was his entire goal.
I'll change out the last example with a different phrase for clarity:
If you said this, a listener might be confused. They might ask "Did he want the degree?" or "Did he have another reason for going?" Maybe Fred's main purpose was to party with college students and blow off classes.
Conclusion:
Remember that changing out these phrases for each other will still make sense. It will still be grammatical. But it will connotate a different mindset on the part of the subject.