OK, piece by piece.
Most people say I wish I could, I wish you would.
Can we use I wish I can, I wish you will?
No. *I wish I can and *I wish you will are both ungrammatical sentences. As pointed out.
I'd like to know what the main differences are between the usage of can/will and could/would when wishing.
Simple enough. It's the interaction of two syntactic and semantic phenomena:
1. The presupposition of a counterfactual complement for the English verb wish
2. The various senses of the modal auxiliaries can, could, will, and would.
The first one means that part of the meaning of wish is that its direct object must not be true;
one can't wish to own something one already owns, for instance. Wishes require counterfactual (aka irrealis) complements. That's why they're linked up with negatives so often.
All predicates have specific subcategorizations like this; requirements for its subject, object, and context. It's part of the meaning of a predicate. Wish, for instance, can take a tensed complement clause, though the that complementizer is often omitted, especially in short or fixed phrases.
A that-complement of wish indicates its counterfactual nature by using a past tense form when referring to the present time
- I wish that they were not in this room.
- I wish they weren't here.
and a past perfect construction when referring to a past time
- I wish that I had not seen the accident.
- I wish I hadn't seen it.
Wish can also take an infinitive complement with A-Equi. This is a very formal construction, and can only occur with a future event.
- I wish to ride in the first class compartment.
But those are the only complement clause types that wish can take; embedded questions and gerund clauses are ungrammatical with wish
- *I wish what I had(n't) seen.
- *I wish (my) (not) having seen it
The other interaction is from various senses of these four modal auxiliaries. All modals, including modal auxiliary verbs, have irregular syntax, ambiguous semantics, and arbitrary pragmatics.
In particular, will, would, can, and could all have multiple meanings. One type that they share in pairs is root variation:
can alternates with could, and will alternates with would.
Historically, this is because in Old and Middle English, modal auxiliary verbs were inflected, and had present, past, infinitive, and participle forms, unlike modern English modal auxiliaries, which are completely defective and only occur as auxiliary verbs.
In this case, will comes from the present tense form and would from the past; ditto can (originally present) and could (originally past). Most of the time, there is nothing "past" about could or would. On the contrary, they usually refer to something in the future (which is also "irrealis", since it's not real yet), rather than the past. But this old "past" sense still shows up with certain constructions, for instance:
- When I was young, I could do 100 pullups, but now I can only do 90.
- When I saw them, they would cook three meals a day, but now they'll very seldom cook.
And they can, in certain contexts, refer to a past that never occurred.
And, as it turns out, the complement clause of wish is one of those certain contexts in which the "past" sense of would and could comes in handy to signal the irrealis status of the complement of wish.
Is it related to the subjunctive? Because one says "I wish I were", not "I wish I am", so are would and could the subjunctive forms of can and will?
It's distantly related to the subjunctive. About second-cousin-once-removed distantly related. There's something like it in the German subjunctive, but German has inflected modals and supports a full assembly of subjunctive verb forms for every verb, while no English verb has a specifically subjunctive verb form; nor are there any subjunctive mood paradigms. Giving the intersection of two rules a name you're not sure of won't really explain anything, so don't worry about "subjunctive". Think of this as one more little foothold on our way up the syntactic mountain.