It's important to use Ę, because it might change the meaning of the sentence: pracujĘ(I work) and pracujE(he/she/it works).
Is there any difference in pronunciation of these words?
Not in the audio provided by Duolingo.
I wrote a bit about the pronounciation of ę today here https://www.duolingo.com/comment/12458434
Why is it not lekarzem?
because it is a different structure. What you are thinking about is "Jestem lekarzem"= I am a doctor, you ussually need instrumental case after "jestem". Here you have Pracuję jako- and you use nominative "lekarz".
in the sentence: "ona pracuje jako modelka " is "modelKA" a nominative femenine?
yes. "modelka" is feminine nominative. (and feminine -a ending nouns, end with -a only in singular nominative)
I think of it is saying " jestę ( -I am -of the doctor breed, classification etc i.e. belonging to, with of being the important word
This is a really really bad recording.
Anyone else get mixed up with 'lekarz' and 'talerz'? Thought this said 'i work as a plate'
No, but thanks for the laugh :)
If I'm a cook, but I'm a perfectionist so I say "but I work like a doctor" do I still say it that way?
"like a doctor" is almost the same, you just remove one letter: "jak lekarz".
Interesting, I must remember that.
Does this sound weird if a woman would say this? Or is "Pracuję jako lekarka" more correct?
With many professions it's rather a subjective choice. Both "lekarz" and "lekarka" sound natural to my ear for a woman.