Learn English – Is it polite to say “what do you mean by ….”

You are right about context and intonation playing a very important role in these sentences.

If you ask a taxi driver to hurry because you are in hurry by saying "Will you hurry?" you are actually asking the driver's willingness and will hurt his/her feeling.

If you are stuck in traffic, asking the driver if he will hurry may indeed hurt their feelings, since it seems obvious that their willingness to hurry is not questioned by the fact that they are simply unable to hurry. (Albeit that some drivers have very original ways to enable them to hurry even in very busy traffic, and that their willingness to do so may increase after the promise of an extra financial reward...)

Indeed, the use of "will" normally inquires directly about a person's willingness to perform a task, and we normally assume that the person is able to do so. Asking them if they are willing when they are obviously unable might be insulting. On the other hand, you could use "would" for a hypothetical situation: "If you knew Spanish, would you help me with my homework?"

"Could you ~?" is usually more polite than "Can you ~?" but only when you ask for help "Can you help me?" is more considerate than "Could you help me?" because when the answer is no the person feels more easier to say no.

"Could" is indeed usually more polite indeed. Where "can" simply asks whether the person is capable of doing something (and implying that you would appreciate they did it!), when you use "could", you are implying that they have to also have a willingness to do it.

So indeed "could you provide an example?" is more polite than "can you provide an example?", but can is the correct form to use when you are genuinely wondering if the person is able to do something:

Can you come to the dinner party this evening? => Are you able to make it, or have you another appointment?
Could you come to the dinner party this evening? => It would be much appreciated if you would come.

"Would you mind ~?" is not polite way of asking but it's rather cynical.

It is actually a very polite way of asking. "Would you mind giving me a hand?" is more polite than any of the previous forms.
However, exactly because is is so polite, it is often too polite in most circumstances! And because of that, it can be very effectively used in a cynical way, in the same way we can use formal forms of address or other (extremely) formal language.

As such, nobody will think of cynicism if a lady asks a stranger:

Would you mind helping me cross the street?

But if I address a friend of mine in this way:

Miss Jones, my dearest, smartest friend, would you mind terribly if I were to decline your kind invitation to your extremely interesting lecture on the history of quilting in 1970's rural Northumbria?

I am quite sure she, and anyone who heard it, would understand that I might be a bit cynical about the interest in the subject as well as my actual appreciation of the invitation - and most importantly, it would be understood I am not actually inquiring whether she would mind my rejection, I am assuming my rejecting should come as no surprise to her!

Answering only your question... It is considered antiquated or overly-formal by most people. It may also be taken in an ironic sense.

Word Origin of Pray from OALD

Middle English (in the sense ‘ask earnestly’): from Old French preier, from late Latin precare, alteration of Latin precari ‘entreat’.

e.g "Pray, continue."

On the other hand, it may be perfectly acceptable to use "beg to request" in India. Usage depends on the context and environment; it's just not in standard usage in the West.