Having the same problem here. Any thoughts?

bezahlen is "to pay (for an item or service)"
zahlen is "to pay (an amount)"

The direct object of bezahlen is the thing you are paying for. There is no preposition needed, as there is in English:
Er bezahlt die Zeitung
Wir bezahlen das Abendessen
Ich bezahle die Rechnung

If you put in a dative object, it's to say who you are paying it to:
Er bezahlt dem Verkäufer die Zeitung

However, you can also use the person as the direct object. In this case, you need the preposition für to say what it's for:
Er bezahlt den Verkäufer für die Zeitung

The object of zahlen is the amount you pay. Here, you can add the purpose of the payment with the preposition für, like in English:
Er zahlt zwei Euro
Er zahlt zwei Euro für die Zeitung

In common usage amongst German native speakers, this distinction is not always made.

bezahlen + dative + [currency] = to pay a sum of money (or e.g. an amount of camels) to somebody: "Er bezahlt der Frau (dat.) das Geld (acc.)" = "He pays the money to the woman". You need a direct object (acc.) here; just "Er bezahlt der Frau" doesn't work. ...

...For that, you'd have to say, "Er bezahlt die Frau (acc.) [für die Blumen (acc)]" = he pays the woman [for the flowers] (i.e., he buys flowers from the woman)

(If the woman is buying flowers for herself, but he offers to pay for them on her behalf, it's "Er bezahlt der Frau (dat.) die Blumen (acc.)" = he pays for the woman's flowers. So, by the way, you couldn't know if "Er bezahlt der Frau hundert Blumen" means "He pays for the 100 flowers the woman is buying" or "He gives the woman 100 flowers in exchange for the 100 apples she gives him".)

"Er bezahlt die Zeitung" and "Er bezahlt für die Zeitung" are interchangeable. Normally you'd simply prefer "Er bezahlt die Zeitung". Besides that, I'd use "Er bezahlt für die Zeitung" to say that "he doesn't steal it / he doesn't get it for free".

What can I say, their so mean! Im sure your very much correct

The way you put it it's wrong, but I think what your German friend was aiming at is: "bezahlt" (derived from the verb "bezahlen") is both third person singular present tense = "[he/she/it] pays" and past participle = "paid".

The past participle for "zahlen" is "gezahlt": "Ich habe / er hat 100 € dafür gezahlt" = "I / he paid 100 € for it" (present tense: "ich zahle, er zahlt")

"Ich habe bezahlt, er hat bezahlt" = "I [have] paid, he [has] paid" (present tense: "ich bezahle, er bezahlt")

"Das Essen wurde bereits bezahlt" = "The food is/was already paid for"

"Der Klempner wird bezahlt" = "The plumber is [being] paid"

"Er bezahlt die Rechnung / den Klempner" = "He pays the bill / the plumber"

I'm not a native German speaker, but I do not think it's a matter of tense. As someone else has written, "Er zahlt" means "he pays" in the sense of paying a bill, or settling an account, and needs an amount after it. "Er bezahlt...." means "he pays...." for something (as here) or he pays someone. But both are present tense.

Umlauts are important here: "zählen" means "to count", "zahlen" means "to pay".

As for "Er bezahlt für die Zeitung", please see the other comments - it's not wrong but apparently not accepted by Duolingo.

"Es geht" can mean: "It (e.g. the cat) goes/walks", "It works out, it's all right the way it is" (e.g. I don't need help carrying this cupboard, I don't need a bag for the thing I'm buying, it's possible to carry the sofa up the twisting staircase), "I'm doing so-so" (as an answer to "How are you?"). As for "Die Maschine geht" ("The machine is working = not out of of order") - I think it's more commonly used in the negative: "Die Maschine geht nicht mehr" = "The machine is not working anymore".

Apart from "es arbeitet" = "Das Kind/Pferd arbeitet", it would only work if you meant "Das Holz arbeitet" = wood expands/contracts/twists when it gets older or when e.g. the temperature changes. "Die Maschine arbeitet" isn't used; you'd say "Die Maschine läuft" for "The machine is working (it's not inactive, it's doing its thing)", or "Die Maschine funktioniert" (it's not out of order).

First Question: As far as I know, "arbeiten" refers to work in the sense of labour, not in the sense of an idea or a machine functioning well. And "Es geht" means that it moves from one place to another., not that it functions well.

Second Question: If you use, "bezahlt", the n "für " is not needed. If you use "zahlt" it is. So the translation should either be, "Er bezahlt die Zeitung" or "Er zahlt für die Zeitung."

Because the root verb bezahlen means "to pay for" so für is not needed technically.