Neither of the current answers sit well with me at the moment, so I'm going to risk adding to the confusion by posting another.
Question 1 (grammar)
First, let's clarify the two verbs in question:
解く solve (a problem)
解ける resolve (itself)
(These are not the only definitions, but for the sake of brevity and on-topic-ness we'll go with these.)
The difference lies in the subjects. With 解く, the subject is the person (or agent) who actually and intentionally does the solving, but with 解ける, the subject is the problem itself, and no agent is given:
私が問題を解いた。 I solved the problem.
問題が解けた。 The problem was solved (solved itself).
Note that 問題が解けた could also mean "[I] was able to solve the problem", since 解けた is both the past potential form of 解く (解く→解ける→解けた) and the past form of 解ける (解ける→解けた).
This shows why the potential form of 解く (解ける) is valid, but the potential form of 解ける (解けられる) is not: 解けられる personifies an inanimate subject. (sawa says this is ungrammatical; I don't know if that's the right term, since it's a syntactically valid form of 解ける, but it's certainly not used.)
○ （私は）この問題が解けない。 [I] can't solve this problem.
× この問題は解けられない。 This problem can't solve itself.
The second sentence might sound fine at first (from a grammatical standpoint), but when you think about it, assigning an ability to an inanimate subject doesn't work here. (Even in English, the better way to express the idea behind sentence #2 is, "This problem will not solve itself.")
Because of this, only 解けない問題 is correct.
Questions 2 and 3
This is an issue of context, as sawa and istrasci mentioned. For example:
私が解けない問題 a problem I can't solve
だれでも解けない問題 a problem no one can solve
istrasci also mentioned the ～にくい suffix. I was about to agree with this, but then I saw sawa's edit which reminded me that 解く can be read as both とく and ほどく, and 解ける can be read as both とける and ほどける. (The meanings differ between readings.) ～にくい can be attached to both ほどく and ほどける, such as in this way:
ほどきにくいくつひも a shoelace that is difficult to untie
ほどけにくいくつひも a shoelace that won't come undone easily
…and to とく, but not to とける:
○ ときにくい問題 a problem that's difficult to solve
However, the ～にくい suffix does not convey the same level of "impossibility" as だれでも解けない does.
Bonus (more "unsolvable")
Generally, once you start throwing out kanji compounds like 解決 and 不可能, you're in the realm of written Japanese, but these might be good to know:
解決できない問題 a problem that (subject) can't solve
解決不可能な問題 an unsolvable problem
解決不可能と思われる問題 a problem thought to be unsolvable