[Snork]

Madoka is not a person, is a character, and you said it yourself, her will to save and help others is not ordinary, and so, it needs a more clear and greater reason that just "the satisfaction of knowing that they helped other people". That's just how you write a character: a clear and fitting motivation makes them easier to relate to, more human.

The difference in opinions here is related to whether we see the provided info sufficient or not. Wanting to be helpful is not something you wouldn't want to bring your daughter up like. And I'm pretty sure that in Japanese society, such a trait is more than welcomed, thus becomes a good element in upbringing. It was apparently successful with Madoka, as we can judge by Junko's words in ep 6. "Loving family" is not just a concept to make you kind and eager to help per se - it's never too far-fetched to assume that a loving family tries to raise the children accordingly.
Now, wanting to be helpful is one thing, but if you're an average girl with no special talents, not enrolling into any clubs (and having Hitomi's image to remind you that you ought to - after all, it's another thing that Japanese schools try to advertise among students) and the most you can do around is take people to the nurse's office... Might make you wonder if the desire to be helpful really matches your potential at all. While I can't say Madoka is desperately looking for a purpose in life (neither was Nanoha, at least before the movie manga retconned her character, and certainly niether was Sakura who seemed to have quite a satisfying life of a teenage girl), but she does express some melancholy about whether she might just live her life away and die. Her self-esteem is not too low, but it's something many people her age come to wonder, if occasionally (and Madoka doesn't seem to be bothered by it ALL THE TIME).
Then, an opportunity rises. There are actually horrible creatures around, threatening people (and there's no guarantee that one of your friends/relatives won't get hit someday - which is proven quite bluntly in episode 4 with Hitomi). Heck, Madoka herself could fall prey to a witch, along with her friend, if not for... MAMI-SAN! There are girls who fight witches and thereby REDEFINE helpful in Madoka's eyes. They do it elegantly and magnificently (check out Madoka's expression when she watches Mami fighting). And here's the juicy part: you, an average girl brought up well and wanting to be helpful to people around but not seeing any prospects in that field, suddenly learn that you can become a Puella Magi! As cool and helpful as Mami-san and presumably just as strong at the very least, if Kyuubey's estimation is true. The ending of episode 2 pretty much nails it down: an innocent, apparently nice office lady (who most likely has someone to grieve over her death) nearly commits suicide she had never even cosidered before, and then she's saved by magic, and here she is, sobbing into Mami's shoulder... You can do the same to people, save their lives - all for free and with a thrown-in bonus of a wish granted magically to you. Who wouldn't AT LEAST CONSIDER this prospect, regardless of upbringing and personality? And for Madoka, it sure looks like a win-win situation. If she wasn't at a loss about a wish, she'd probably become a Puella Magi right way - as she did in previous timelines. Save the poor cute cat whom nothing else will save anyway and proceed to save people whom, likewise, nobody but the likes of you would sufficiently protect from witches - sure, why not? And in TL4, preventing destruction, grief and your classmate getting killed before your eyes, when nothing else is stated to do that and you can do it? COUNT ME IN ALREADY!

So, to summarize.
- initially Madoka is what Junko describes her as - a good and kind girl. Her wish to be helpful is moderate, partly restrained by her lack of belief in her abilities (otherwise, she'd have been a triple scout already ) and not too uncommon for a person who apparently was brought up with such an wish to keep in mind.
- her wish to be a hero directly derives from the seeming opportunity to become one with a flash spark. The wish to help people suddenly gets a definite prospect sketched out - and it's more than Madoka has EVER hoped to be able to do. You won't be able to explain her eagerness to fight witches unless you consider HOW she views the system at the time (and in timeline 1 she herself becomes a poster child for it together with Mami, in Homura's eyes).
- her self-denial and self-sacrificing, while they MAY be based on her personality and the first two points above, do NOT look like her innate traits to me. If anything, Madoka knows well what Junko voiced in ep 11: her life does not belong to her alone, and her family will be in great pain. That's partly why she turns down Kyuubey's offer in ep 4 after witnessing and painfully realizing the old "Anyone Can Die" trope. Up to this point, there is no self-denying Madoka. However, if this girl has a heart big enough to sympathize with other people and want to help them, what to expect when people dear to her start getting involved deeply? When Hitomi's endangered, she does her best to save her. When Sayaka contracts, she insists on tagging along because she can't leave her friend alone in situations she has seen to end tragically. And tagging along as a helpless spectator makes her consider contracting again, more and more. She's hit again with the episode 6 revelation, but then it's seeing Sayaka suffering that makes her as desperate as to put her own soul on the line - because Sayaka is not even some abstract "people to help", she's her dear friend whose life is being ruined rapidly... and Kyuubey keeps hinting Madoka could change it. As I see it, THIS is where the Madoka we've seen at the beginning of episode 12 comes from. Her self-denial and sacrifice does not lie in her character that needs backstory motivation - it's her character development that starts from ability to sympathize with others and a simple wish to be helpful (not necessarily to a Spiderman extent and not necessarily bordering on inferiority complex), then goes along the line of the events Madoka experiences.

What is this I don't even...

Jokes aside (but only sort of), seriously, I'm starting to lose sight of why people have so little love for Sayaka. Really, why? Maybe I just look at her differently than you do, but I can't think of a single thing to truly condemn her for. As far as I can tell she's nothing more than a perfectly fine character that got hit psychologically one too many times; that's not really her fault, though, so I fail to see how that might affect somebody's perception of her.

And on your previous point of her surviving had she accepted Homura's Grief Seed, I doubt it. Episode 11 showed just how easily the Soul Gem can become tainted by grief; with Sayaka in the state she was in, her Soul Gem would probably be just as dark again in a day or two. Grief Seed or no, she was on a downward spiral that wasn't likely to change.

What Sayaka needed was not a grief seed, but a reason to live. She entered a heavy BSoD and her only fault ever is turning down all attempts to snap her out of it. She IS a tragic character, but I don't see how people can be annoyed with her. Especially the reviewers who usually complain bout characters being not true to life - now, Sayaka's behaviour is QUITE believable for an energetic, naive and lovesick teenager thrown into the aforementioned circumstances. She meets her end because she failed to get a hold of herself and because the plot (unlike many other plots in anime, and especially in mahou shoujo) did NOTHING to bail her out of it by some external means. Just as Urobuchi admitted to see things - "leave them on their own and they'll naturally develop in the worst way possible".
And I think Homura knew this. But she tried to postpone Sayaka's fall at least, because she desperately needed to do something about the situation before it led to Madoka's contracting.
Personally, I love Sayaka and I felt like my heart was being slowly driven over by a paver while watching her arc. She deserves a better fate - yet the most she can get now is being aware of changing Kyousuke's life for the better and maybe finally hitting it off with Kyouko after she joins her in Madoka's domain. Rest in peace, our genki knight.

And she instantly gave up on her love just because Hitomi said she will confess...

She gave up on her LIFE, let alone her love. Since episode 6 she views herself as a walking corpse - Shinji Ikari's self-esteem is over 9000 compared to this. I'm afraid it wouldn't matter if Hitomi gave her a day or a week, unless Sayaka herself got over her real problems in that period. Of course, this 24 hours ultimatum just added oil to the fire. Sure, the show's one-cour pacing benefited from that, but Sayaka obviously didn't...