If I worked the phone lines at the Egyptian museum, and someone asked me:
When are you open?
or:
What time are you open?
Then I would give them the full range of the hours the museum is open:
We are open from 9 to 5.
However, if they asked me instead:
What time do you open? (or "When do you open?")
Then I would give them the time the museum opens:
We open at 9 AM.
I'm guessing that 5 o'clock is when the museum closes, not when it opens. So, I'd expect that dialog to go something like this:
What time are you open until?
or:
When do you close?
In which case I'd say:
We close at 5 o'clock.
The answer:
We are open at 5 o'clock.
tells me that's the time when the museum opens its doors. Five in the morning seems like a good time for a breakfast diner to open, and 5PM might be a good opening time for a nightclub, but it's hard to imagine a museum opening at five o'clock.
However, had your question read:
We are open until 5 o'clock.
that would make more sense.
Are you open today? - This is like saying "Are you open right now?"
Do you open today? - Sounds like "Will you open today?"
They're both pretty interchangeable, however if I called a business to find out if they were open, I'd almost certainly ask the first question.
Best Answer
The reason the person was confused was because you started your question with:
That is the start of a yes-or-no question, such as:
If you are asking for the time, you could have asked instead:
The "How late are you..." opening of the question lets the listener know that you will be asking for the time, and not wanting a yes-or-no answer.