Would you mind providing some details, please?
Is perhaps the more 'correct' and slightly more formal way of rephrasing your question. Or the example given by @unorthodox grammar is just as good, and slightly less formal:
If you don't mind, would you please provide some details...?
The please is optional in either place, as the 'would you mind?' conveys enough politeness in my opinion, but it wouldn't hurt to add it in. It would usually come immediately before the action that you are asking the other person to do, i.e. please provide... or tagged on to the very end of the question
If you have quite a long sentence, the 'please' can be placed at the end of the clause where the actual request is made – to avoid waiting for the end of a very long sentence:
Would you mind providing some details please, for us to capture the steps required for future requests?
Alternatively:
Would you mind? Could you (please) provide some details...?
Is a bit more conversational. (Not something you would write in a formal email). Both of the examples you gave I have seen written or heard in everyday conversation, and yes, they do look/sound slightly odd constructions.
If you are making the request in an imperative, but still polite manner, the 'please' can come at the beginning of the request:
Please provide some details for us to...
In highly traditional English, will/would in the first person is/was not used to express possibility only, but rather intent of some sort. Even so, the large majority of modern writers and speakers have deviated from this practice and use will/would for a neutral sense of possibility in the first person too.
At any rate, will/would has been used by the majority to express possibility for the second and third person for many centuries; it has been used as in your examples for a long time by almost everyone. There is therefore not even the hint of an issue or problem in your examples, and so there is no reason to remove the will/would.
Best Answer
The contracted forms (2 and 3) are typical of speech and informal writing. Some contractions are found in formal writing, but not, on the whole, the contracted forms of have. That apart, all three are grammatical and mean much the same thing, but with slight differences of emphasis. In speech, the differences can be signalled by changes in stress. Which is chosen will depend on what has gone on previously in the conversation.