[영어Q&A] Do you know vs Do you happen to know

A1.

No difference at all. They both ask the same question. This is more about a style rather than a correct usage question.

English people would tend to say, 'do you know' whereas Americans are more likely to say, 'do you happen to know'.

A2.

The use of 'do you happen to know' would not be used to pass on information, but to enquire after information that the other person may not, in fact, have:

"Do you happen to know/would you happen to know what bus** will take me to the town centre?"

(Brit.Eng.)

** A person in Britain would understand that 'what bus' refers to 'the number of the bus'. The number is displayed at the front and side of the bus, and indicates the route that the bus takes.
The reply might be, "Either a number 3 or a number 14 will take you there."

A3.

'Do you know that Silver was arrested?' is ok but I would say 'did you know that Silver was arrested"

'Do you happen to know that Silver was arrested' is not correct.
'do you happen to know IF Silver was arrested' is the correct way to ask.

A4.

I am not sure if any generalizations about British/US English can be drawn here. 

If asking a question, it would be more idiomatic to ask in the past tense: "Did you know that Silver was arrested?"

Present tense (Do you know...) might be used to ask more forcefully.

A5.

"Happen to" refers to chance, not obligation or need.

Do you happen to know = "do you perchance know", "might you know", "would you know".
Did you know = "have you heard".

A6.

Not here. As mentioned above, Do you happen to know that... is not correct here.

As MilkyBarKid said, that construction is used to seek information, hence why Do you happen to know if/when/how/where/who/etc. is used, but not that, since that does not introduce an indirect question (which seeks info), but rather a statement (which states info).

And I'll add that it's a sort of "distancing" mechanism, similar to "Could you pass the salt?" instead of "Will you pass the salt?" It conveys a sense of politeness.

A7.

I think there's usually a difference between the meanings of Do you know that...? and Did you know that...? I'm not at all happy with the suggestion that the one is 'more idiomatic' than the other.

Did you know that X? can mean I know that X, but do you know it too? It's very much the same as Have you heard that X? In this meaning, I agree that Do you know that X? could be asking the same question, but would be a less likely form of words.

Do you know that X?, on the other hand, is what you say when you wish to check on the truth of what the other person is saying or has said to you. The stress would be on the knowand the question means Are you certain that X? The suggestion is that the other person has told you that X, but you aren't sure he's right. You want to be sure, so you say Do you knowthat X?

Did you know that X? would be wrong in such circumstances, though it could, of course, be asking about the other person's state of certainty in the past.

These questions can become important when you are dealing with people who use the word know lightly.

Do you happen to know that X? is not a possible question in BE as I know it to mean Have you heard that X? I'd say Do you happen to know if X? if I wanted to know if the other person could tell me if X? Do you happen to know if the Manchester train leaves from this platform? would be a polite way of asking if the train left from this platform.