Word Usage – Differences Between ‘As Usual’, ‘As Ever’, and ‘As Always’

Native speaker here – American English (New England, Boston area).

I may be grasping at phantoms here, but I have a very faint impression that when one uses "would always", it implies "on those occasions on which that person would do this thing, they always elected to do it in the following way", while, in contrast, "was always" implies that the person (or thing) so described was multiplying occasions to do the thing.

That is, using your examples,

She would always send me strange birthday gifts.

Suggests to me that on the occasion of your birthday, she could be relied upon to choose gifts that were strange, while

She was always sending me strange birthday gifts.

would suggest that she wasn't waiting for your birthday to send those strange dirthday gifts!

Likewise,

Sam and Mary would always choose the most exotic vacation destinations.

suggests that Sam and Mary's vacation destination choices were consistently exotic, while

Sam and Mary were always choosing the most exotic vacation destinations.

suggests that Sam and Mary have a very, very busy vacation calendar.

This

Ned would always show up at our house without calling first.

suggests that it was Ned's custom to drop by without calling first, while

Ned was always showing up at our house without calling first.

suggests Ned is making a serious nuisance of himself by imposing on us so very frequently. (Note that in this example, the "was always" formulation has a much strong negative valence than the comparative "would always.)

The difference, I think, is that used this was "was always" is hyperbole, while "would always" is merely a generalization.

Your description of "was always" + verb as implying a negative is correct, but it doesn't explain how that works. It is using hyperbole, which is a figure of speech which conveys valence by exaggeration. When someone says of someone "he was always doing that thing!" it is not usually meant literally; it is meant as exaggeration for effect.

In contrast, "would always" may well be meant literally, or close to literally. When somebody says "he would always do that thing!" they may well mean "whenever he did a thing, it would be that thing!"

By way of illustration: I had a friend who would always be late to parties. Famously so. Like, numerous times he arrived at parties when the last of the rest of the guests were saying good-bye to the hosts. Once, story has it, he showed up the day after the party. Yet I wouldn't say of him that he was always being late to parties because he didn't go to parties all that often. It wasn't like his life was full of party-going, only late. But, by gum, when he went to parties, he would be late; you could bank on it. So my saying of him that he would always be late to parties was perfectly, incontestably, literally true.

I shall be a great deal too far off to trouble myself about you.

Translation, chunk by chunk:

[ I shall be ] [ a great deal ] [ too far off ] [ to trouble myself about you. ]
= [ I will be ] [ very ] [ too far away from you ] [ to do anything about you. ]
~ "I will be too very far away from you to do anything about you, my feet."

A great deal is a set phrase, meaning: "a large amount or quantity of something".

Why did Alice say so?

Because she was stretched out, like a telescope (see the image below), by the power of the cake after eating a cake in the previous chapter. (The sentence in question is from the first paragraph of chapter II of Alice's Adventures in Wonderland.)

^{Source: https://www.cs.cmu.edu/~rgs/alice-II.html}