Learn Hardcore Icelandic: Ég get ekki beðið lengur.

get is the 1st person singular present tense of the verb geta (to be able to / can).
So Ég get literally means I can / I am able to.

Why is it Ég and not some other form of I/me?

Ég is the nominative (subject) form of I in Icelandic. Since Ég is doing the verb get, it stays in the subject case.

Why is the negation ekki placed after get?

In Icelandic, ekki typically comes after the finite verb (the verb that’s conjugated for person/tense).
So you get the common pattern: Subject + finite verb + ekki + rest
→ Ég + get + ekki + beðið + lengur

Why does Icelandic use beðið instead of an infinitive like bíða?

With geta (and several other verbs), Icelandic often uses the supine form (which looks like a neuter past participle) rather than the infinitive.

infinitive: bíða = to wait
supine/past participle form used here: beðið

So get beðið is the normal “can + verb” structure here.

Is beðið a past tense? It looks like waited.

Not in meaning here. beðið is formally a past participle/supine form, but in this construction it doesn’t make the sentence past. The time reference comes from get (present).
So the sentence is present: I can’t wait any longer.

What verb is beðið from, and why does it change so much from bíða?

It comes from the strong verb bíða (to wait). Strong verbs often change the vowel in different forms.
A simplified set of forms is:

bíða (infinitive)
bei… (past forms)
beðið (past participle/supine)

That vowel change is a normal strong-verb pattern in Icelandic.

What does lengur mean exactly, and why not just lengi?

lengur means longer / any longer and is the comparative form related to lengi (for a long time).
In negative sentences, lengur is very common for any longer:

ekki … lengur = not … any longer

Could I change the word order, like Ég get beðið ekki lengur?

That word order is generally not natural in modern Icelandic. The standard placement is:

Ég get ekki beðið lengur.

You can move things around for special emphasis in some contexts, but as a learner it’s best to keep ekki right after the finite verb (get).

Does this sentence mean “I’m excited and can’t wait,” or “I’m unable to wait”?

It can cover both, depending on context, but the most common everyday sense is I can’t wait any longer in the “I can’t keep waiting” sense (impatience / it’s taking too long).
If you want the clearly “excited” sense, Icelandic often uses expressions like:

Ég get varla beðið! = I can hardly wait!

How do you pronounce Ég get ekki beðið lengur?

A learner-friendly approximation:

Ég ≈ yeh(g) (the g is soft/voiced; often not strongly pronounced)
get ≈ get (short vowel)
ekki ≈ ehk-ih (with a “k” sound in the middle)
beðið ≈ BEY-thith (the ð is like the th in this)
lengur ≈ LEN-gur (with g fairly soft; ur like a relaxed ur)

(Exact pronunciation varies by speaker and region, but that will get you close.)