A three-part verb

Question

I am a verb; I come in three parts.

The first recalls a pirate.

The second reminds me of you!

The last could be the opposite of out.

-

These parts, when combined but not whole,

Result in what sounds like a question.

Though don’t tell me teacher that it ended in a preposition

-

If you consider my case in a lower court,

You can identify my parts in other ways, too:

The first is a part of the last;

The second is the inverse of the last

The last is naturally the first plus more,

And the inverse of the middle.

-

And now for my whole!

I carry a connotation of swiftness, but that’s not always true.

I can occur slowly, on a computer

Or with a mouse, or a spider, or an ant – relativity is key.

(And not the kind that Einstein discovered).

-

I am sister to some cousins of wordplay, diversion,

A certain nearby star, and completion.

If you haven’t found me yet, don’t run away!

I’ve dropped myself somewhere around here –

Perhaps you can spot me in that way!

What am I?

Tip - the divisions of the lines have nothing to do with the answer; they just made the riddle look nicer.

Answer 1

Partial

I think the answer is

Run

I am a verb; I come in three parts.

Three letters

The first recalls a pirate.

R - Pirates in movies are associated with saying "Aarrr" which sounds like 'r'

The second reminds me of you!

U sounds like you

The last could be the opposite of out.

N sounds like 'in'

-

These parts, when combined but not whole,

Result in what sounds like a question.

R U N sounds like 'are you in?'

Though don’t tell me teacher that it ended in a preposition

'in' is a preposition.

-

If you consider my case in a lower court,

lower case, as suggested by GentlePurpleRain

You can identify my parts in other ways, too:

The first is a part of the last;

The drawing of lowercase 'r' looks like a subset of 'n'

The second is the inverse of the first

On a keyboard they are opposites?

The last is naturally the first plus more,

Again referring to drawing the additional stroke on the 'r' to get 'n'.

And the inverse of the middle.

'n' is 'u' upside-down

-

And now for my whole!

I carry a connotation of swiftness, but that’s not always true.

Run can mean to move one's legs in a swift manner

I can occur slowly, on a computer

You can run a computer program which doesn't necessarily need to be quick

Or with a mouse, or a spider, or an ant – relativity is key.

Different sized creatures will run at different speeds

(And not the kind that Einstein discovered).

-

I am sister to some cousins of wordplay, diversion,

A certain large star, and completion.

If you haven’t found me yet, don’t run away!

Explicit use of the word run

I’ve dropped myself somewhere around here –

In the previous sentence

Perhaps you can spot me in that way!

Answer 2

I think the verb is

"run"

First section:

r - recalls a pirate's "arrr"
u - reminds me of you, because its a homonym
n - another homonym, "in" could be the opposite of out

Second section:

With all letters in lower case
To make the letter "r" you make most of the letter "n"

I think there's a typo here and it should read "the second is the
inverse of the last, as "u" is an upside down "n"

Third section:

To run indicates quickness, but a computer could run slowly and
a mouse, a spider, and an ant would all run at different speeds

Fourth section:

I take "sister to some cousins" to mean the word rhymes with words
that have these meanings

wordplay = pun
diversion = fun
large star = sun
completion = done

and the word "run" is hidden in this section