Is "would" the past future tense of "will" or just a modal verb?

Would can be either conditional or subjunctive, but it is often used (as your examples demonstrate) interchangeably with will.

Will is an inquiry after the consent of the respondent, whose inclinations comprise the sole issue at hand. "Will you give me some advice?" literally means it is up to "your" discretion either to give or withhold the advice.

"Would you give me some advice?" on the other hand implies some other conditions may affect your decision. Unstated but implied in this sentence may be some other information. Or there may be a contextual linkage or even a direct statement. I think of Dr. Seuss's Green Eggs and Ham:

Would you eat them in a box? Would you eat them with a fox?

The terms of the conditional are clearly set forth. The questioner is proffering various inducements to sweeten the deal for the recalcitrant hater of "Sam-I-Am".

In any case, asking either question in conversation will, in the vast majority of cases, be understood simply as a request for advice, without all the grammatical analysis.

Conventionally, would can be either the past simple or the past subjunctive of will. To put it very simply, it is best considered past simple if you are looking forward from a past perspective in a story. In that case, it is used in the middle of a narrative that is in the past tense. Both the narrative and would are then simply a description of the past; what would be will in a narrative in the present tense becomes would:

• She said she would succeed. (In direct speech it would be: She said: "I will succeed".)

• He knew they would find him eventually. (In direct speech it would be: He thought: "I know they will find me eventually".)

In most other cases, it is past subjunctive. This tense can be used in several ways, of which the conditional is the most frequent:

• If he were rich, he would still be a bastard. (The conditional is used to express that "he will still be a bastard" is only true if the "if" condition is fulfilled.)

• She would like some more tea. (Here there is some implied condition, such as "if you asked her", "if it were possible", "if she were permitted to speak", "if it weren't rude", etc.; that is why conditional past subjunctives are often used to express politeness.)

All the above generally applies equally to the other modal verbs, can, shall, and may.

If you say "will you give me some advice?", this is a perfectly fine and polite request, though perhaps a tiny bit old fashioned. It could theoretically be a question about the other person's desire, but context makes it clear that this is not what is intended.

"Would you give me some advice?" is an attempt at even greater politeness, because the conditional makes the request even more tentative, as explained above. Take your pick.

Either would or will work, but you wouldn't change the noun:

Would you give me some advice?

and

Will you give me some advice?

Is “would” the past future tense of “will” or just a modal verb?

Would is the past tense of will in sentences like

He said he would be away for a couple of days.
He wanted out, but she wouldn't leave.

The difference between Would you give me some advice? and Will you give me some advice? is that the first is considered a polite way to ask help, while the second (depending on the context) could be also understood as expressing desire, consent, or willingness.

Will you have a cognac?