FlashCards: what is the difference between Est-ce que and pouvez vous

I was doing the FlashCards (By the way RF I love FlashCards).
The question was, "Can you understand me?" I had 2 answers in my head, polite and informal -
Pouvez-vous me comprenez ?
Peux-tu me comprendre ?
The correct was Est-ce que vous me comprenez ?

I understand that "est-ce que" is used to turn a statement into question and can't be translated literally to English. Secondly what happened to the word "can" in the question and lastly when do we use "pouvez-vous and peux-tu" if "est-ce que vous me comprenez" is correct or better answer?

Good one. I agree with you that pouvoir  could also add that "can" (Pouvez-vous/Peux-tu me comprendre ?), but it is a bit superfluous because you could just conjugate comprendre.

I think this is more of an Anglicism because to indicate ability, we add "can", instead of just saying yes or no on the ability.

(Do/Can) you hear me? -> Tu m'entends? 
Yes, I (can) hear you. -> Oui, je t'entends.

Please forgive me for repeating what you just told me. I want to be absolutely sure that I understand what you are teaching me or slowly in snail pace moving towards thinking in French one day. 
 
(1) First mistake was Peux-tu me comprends (I forgot to conjugate but I did with the vous form) Yes?

(2) Tu peux me comprends as well as the vous form, use  a statement as a question if I insist on using the pouvoir but leaving out the "est-ce que". Did I understand you correctly?

(3) Since it will be a while before I can "think in French" I should try to think in texting, as simple and as succinct as possible such as leaving out the "est-ce que" is perfectly alright. 

I had been wearing my former hat for too long. I talked almost like I wrote letters of communication and presentations, just without the technical jargon. Diana-S1 said that (different question but similar idea) she would have counted my answer as correct or good whereas I mark it as so-so because it didn't look anywhere near the answer given. 

merci mille fois

For #1, it's the opposite. It stays in the infinitive because pouvoir is conjugated.

For #2, if I understand the question correctly, can the «Est-ce que» be omitted if you use an inversion (or just even "inflect up") to form the question? Yes.

So the possible answers are:

Highly formal, used in written text
Pouvez-vous me comprendre ? / Peux-tu me comprendre ?
Me comprenez-vous ? / Me comprends-tu ?» (very formal, used in written text)

Less formal than inversion, used in written and spoken French
Est-ce que vous pouvez me comprendre ? / Est-ce que tu peux me comprendre ? Est-ce que vous me comprenez ? / Est-ce que tu me comprends ?

Colloquial/spoken/informal, inflecting up to form question
Vous pouvez me comprendre ? / Tu peux me comprendre ?
Vous me comprenez ? Tu me comprends ?

Thank you thank you!

#1 immediately understood/remembered but had to be reminded, not able to think it through by myself yet. 

#2 great - aside from the same mistakes I made in conjugation! 

Million thanks for writing out all the possible ways.