Learn English – when to use “to” and when not

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!

1) Monday should be fine.

2) Monday would be fine.

Should in sentence 1 is not the same use of should that we find in examples such as:

• What should I do?
• You should apologise.
• You should eat five portions of fruit and vegetables every day.

The sentences above are asking for advice or giving advice. We can think of advice here like a form of weak obligation. If someone gives us advice, there is some pressure on us to do that thing - but we don't have to do it - it is our decision. This kind of meaning, when we talk about obligation and permission, is called DEONTIC modality.

Sentence 1 is NOT about deontic modality. This type of should is about EPISTEMIC modality. Epistemic modality is about knowledge and belief. Think about the following sentences:

• It might be in box 3.
• It should be in box 3.
• It must be in box 3.

The first example above shows that the speaker has a weak belief that it is in box 3. The second sentence shows that the speaker has a fairly strong belief that it is box 3, but she's not certain. The last example shows that the speaker is certain that it is in box 3.

In the Original Poster's example, (1) indicates that the speaker has a strong conviction, a strong belief, that Monday will be fine. Of course this is the technical meaning of what they are saying. The effect of saying this sentence is probably "Yes, choose Monday".

Would in sentence (2) indicates a logical result of choosing Monday. The sentence is like the last part of a conditional:

• If you chose Monday, Monday would be fine.

Here the speaker is definite about the fact that the result of choosing Monday is it's being fine. They aren't indicating any doubt about it. Again the effect in the conversation is probably "Yes, choose Monday".

Hope this is helpful!