gets = "παίρνει" and takes = "λαμβάνει"?
get = παίρνω
receive= λαμβάνω
take = παίρνω, λαμβάνω.
Τhose three are interchangeable in many cases. ^.^
Is φέρνει also fair alternative like 'fetches'?
φέρνει means brings in Greek.
I wrote Ελένη παίρνει το καπέλο της. It says I'm wrong because I did not say η Ελένη παίρνει το καπέλο της. Do people actually talk like this? Is it considered bad Greek to omit that article at the beginning of the sentence?
Do people actually talk like this? Is it considered bad Greek to omit that article at the beginning of the sentence?
Yes and yes.
The definite article is required before a person's name in Greek when you're talking about (rather than to) that person.
μπορώ να πω "είμαι χόλ"; θα είναι γραμματικά σωστό;
You mean to say "I'm in the hall"? It's correct, but very very colloquial, especially if used for rooms.