Actually, I think trying a "hard problem" may be a good idea IF

1) You have a fair evidence that you are strong enough to tackle things other clever people gave up on. The evidence should be tangible. The best evidence is, of course, having solved at least one hard problem already, but that, obviously, cannot be applied to your first hard problem ever. Sometimes a good indication is other people saying something like "You should stop stealing other mathematicians' daily bread and do some real thing that no one else can do!" (Note that you *shouldn't* follow the first part of this advice.)  

2) You have an escape strategy. That may be thinking of something else in parallel, making sure that your plan is such that even a partial progress can be of value, etc.

3) You are not afraid to fail and are used to the feeling of being a hopeless idiot (meaning you can calmly admit this frustrating fact about yourself without any reservations, excuses, or other kinds of self-deceit and still push ahead at your full strength). 

4) You have enough free time and do not care too much of your career ups and downs.

5) You are sufficiently open-minded to see things at unusual angles and are trained to figure out reasonably quickly whether any given idea may possibly work or it certainly won't. Note that both  are tough skills, which are almost completely untouched in most standard treatises on problem solving.

6) You love the problem. This should, actually, be #0 rather than #6, and it is hard to explain what it means in rational terms, but you can feel it when it happens.  

If those conditions are satisfied, go ahead and try shooting the Moon. If not, you'd better make your way up slowly step by step like most of us, picking the fight just slightly bigger than your own size every time. 

I'm not a great believer in "having a new idea from the start". The new idea or a combination of ideas usually comes eventually when working on the problem and the moment it comes is often very near the end of the story. The trail of failures that precedes it is well-hidden, but we all start with "I have no method, no feeling, no tools, no clue, and no hope" and proceed through "twisting this, we can get a bit more or something a bit different, however the main difficulty remains untouched". You have to figure out not only what doesn't work but also how exactly it doesn't work. Most of the time is spent on constructing examples and counterexamples to the steps in your initial plan, digressing into simpler models, checking that no information is lost at each particular step, i.e., that if the original theorem is correct, then the intermediate lemma you want to try is at least very plausible, and so on, and so forth. I do not know how it works for others, but for me any non-trivial problem is a scattered jigsaw puzzle, not an originally blurry but complete picture I merely need to focus the camera on. 

I'm not sure how much credibility I can claim myself when talking like this about solving *hard* problems, but, fortunately, most of these claims aren't my creations: I merely believe they are true and the opposites are false. So, take all this with a healthy grain of salt and keep in mind that out of 100 mathematicians, at most 5 are qualified to shoot the Moon in principle and, out of those 5, at most 1 will score a hit when making this long shot, so don't judge us, professors, too harshly when we just know our limitations and are unwilling to try to jump above our heads. There is a lot of stuff at the knee level that needs to be done and some of us (including myself) just feel that it will be more efficient to spend most of our time doing it there. One becomes a loser not when he aims and shoots lower than the Moon but when he stops seeing it in the sky :). 

As to the formal question list, I would answer as follows:

Should one attack such hard problems at all? 

Yes. The gods won't do it for us, so it'll have to be one of us, poor mortals, who should try.

If one should, why and when? See #1-#6 for "when". As to "why", if one asks this question, one shouldn't.

Will studying hard problems span new ideas? 

Possibly. It can work out either way.

Is it even a necessity to understand some hard problems and, especially, why they are hard to solve? 

No, nothing is absolutely necessary. You can live and work perfectly well without it. 

Or is it just pure waste of time? 

This depends on who and what you are.

Or is it that one should learn some hard problems to educate oneself but not spend time attacking them?

That works for some people too.