"怎么办“ is a set phrase. It sort of means "what should I/we do?" A more literal translation would be more like "how should I proceed now?" This is asking how to navigate a situation or solve a problem. This phrase isn't used to ask something like "what should I do on my day off?" For that, you could use “做什么？”

Apparently, zenme ban is a phrase that means what to do. So it's just a phrase that doesn't mean what it literally would mean, like in English you have for example "what's up" but you're not asking about anything that is up.

办 also means 'to manage' or 'to handle', so maybe you can think of it as 'How can I deal with this situation', 'How can I cope with this?' or 'How should I handle this?'

On another app (Memrise) I am using 怎么办 is translated as "what to do". Perhaps 做 is used with a specific activity whereas 怎么办 is used without one? This is speculation. I have a Living Languages Mandarin program and one of the speakers asks another 你在做什么?

In my opinion the idea of "how" (怎么办) here refers to "in what way should I do", not "to what extent should I do" as you might have mistook.

I have the same doubt

If you look in pleco for 办 one of its examples is almost the same as this: 我该怎办 for "what shall i do".