Learn Hardcore German: Mein Hut liegt noch im Auto; kannst du ihn mir geben? - My hat is still in the car; can you give it to me?

liegen describes a resting position (literally “to lie”). It highlights that the hat is lying somewhere as a physical object: Mein Hut liegt …
sein is neutral and just states presence: Mein Hut ist … For everyday speech, Mein Hut liegt noch im Auto sounds very natural. With movement, use the transitive pair legen: Ich lege den Hut ins Auto.

What exactly does noch add?

noch here means “still.” It implies the situation hasn’t changed yet (the hat remains in the car). Related contrasts:

noch vs. schon: still vs. already (e.g., Er ist noch da / Er ist schon weg.)
noch nicht = not yet
immer noch = still (often stronger emphasis than plain noch)

Why im Auto and not in dem Auto or ins Auto?

im is the standard contraction of in dem.
in is a two-way preposition. With location (no movement), it takes dative: in dem Auto → im Auto.
With movement into something, it takes accusative: ins Auto (contraction of in das Auto).

Why is it ihn and not es?

Because pronouns follow grammatical gender. Hut is masculine (der Hut), so the accusative pronoun is ihn. es is neuter and would not refer to Hut in standard German. So:

nominative: er (der Hut)
accusative: ihn
dative: ihm

Why mir and not mich?

The verb geben governs two objects:

the thing given = direct object (accusative)
the recipient = indirect object (dative) So in Kannst du ihn mir geben?, ihn (the hat) is accusative, and mir (the recipient) is dative. mich would be accusative and is wrong here.

Is the order ihn mir fixed, or can I say mir ihn?

Both are possible, but the default with two personal pronouns is usually accusative before dative:

Neutral/default: Kannst du ihn mir geben?
Marked/emphatic or rhythmic variation: Kannst du mir ihn geben? (often with stress on ihn) A special note about es: it very often comes early, so Kannst du es mir geben? sounds much better than Kannst du mir es geben?

Why is kannst at the beginning and geben at the end?

In yes–no questions, the finite verb goes first. With a modal verb:

finite modal first: Kannst
subject next: du
objects/adverbs in the middle: ihn mir
main verb (infinitive) at the end: geben Compare the statement: Du kannst ihn mir geben.

Could I just say Gib ihn mir (bitte)! instead?

Yes. That’s the imperative and sounds more direct. You can soften it with particles/politeness:

Gib ihn mir bitte!
Gib ihn mir mal, bitte! (colloquial softener mal) Using a modal question (Kannst du …?) is also a common, polite way to make a request.

How do I make it more polite than Kannst du …?

Good options include:

Könntest du ihn mir (bitte) geben? (more polite/softer)
Formal address: Könnten Sie ihn mir (bitte) geben? Adding bitte and/or mal can also soften the tone.

Is the semicolon correct here? Could I use a comma or a period? And should kannst be capitalized?

A semicolon is perfectly fine between two closely related main clauses.
In German, a comma between two main clauses is also acceptable: …, kannst du …
A period is fine too, for a clearer break.
After a semicolon, you do not capitalize kannst (unless it begins with a noun or proper name).

Why mein Hut and not meinen Hut?

Because Hut is the subject in nominative case. The possessive mein inflects:

nominative masculine: mein Hut
accusative masculine: meinen Hut Compare: Mein Hut liegt im Auto. vs. Ich nehme meinen Hut.

Could I use bringen or holen instead of geben?

geben = hand/give to someone (focus on the transfer)
bringen = bring to someone (implies moving it to you)
holen = go get/fetch it (from somewhere) Depending on what you want: Kannst du ihn mir bringen? (bring it to me), Kannst du ihn holen? (go fetch it). Here, geben fits if the person already has access to the hat and you want them to hand it to you.