bezahlen is used when you want to say that you pay for something (mostly with invoices) Ich bezahle die Rechnung. zahlen can be used as a synonym here, but when you want to say that you are paying (here we refer to / emphasize the action of paying) you only use zahlen: Heute zahle ich. "Herr Ober, zahlen bitte!" if you are in doubt, there is a great dictionary called linguee, where you can sentences with translation, and you can get different examples for zahlen / bezahlen, just enter the words and you will see sentences that use those words! Hope this helps, and good luck!

Paying an invoice would be "eine Rechnung bezahlen", but paying an employee would be "einem Angestellten seinen Lohn zahlen", so depending on the situation you might have to use one or the other because it is tied to a certain expression, but most of the you can use them like synonyms.

yes the be- prefix seems to relate to an accusative object. also genitive object.

There's basically no difference. I don't think there's a rule. But there are different situations where you use one or the other. "bezahlen" is the more common one. "zahlen" is often used in gastronomy. For example: "Ich möchte zahlen", "Ich zahle auch für sein Bier". But in conclusion: both are synonyms