Word Choice – Differences Between ‘In Order To’, ‘So That’, and ‘So As To’

I personally use the US when speaking about the country generally, the States when talking to someone overseas (about coming over, comparatively), and rarely, or never the USA.

However, I think generally, most people use the US in conversation. I think the only time I hear people use the USA is when they say it emphatically and patriotically.

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

1. in a way that
2. in such a way that
3. in this way
4. in doing so
5. in order to
6. in order that
7. so that
8. 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.