Yes, Vlad, Alucard presumably gained Schrodinger's powers.
When he returns to Integra at the end of the series, he states that he's been killing off all the souls inside him, save for one. It's a safe bet that that final soul is that of Schrodinger (otherwise Alucard would probably not still "exist"), and with that being the case, he would most likely have gained his powers. As such, he would no longer need the millions of souls to be "immortal," he would only need Schrodinger's to ensure his immortality.
As for the discussion about his defeat as Abraham and crew's hands being genuine, and his "only a man can kill a monster" lines, it's explained fairly well in little perks of his in both manga and shows.
Alucard views mankind as beautiful in both their ability to age (seen in many instances where he refers to old age and the act of aging as beautiful, such as when Walter betrays Hellsing and becomes young again; he tells him that his old, aged body was a thousand times more beautiful) and in their mortality by means of mindset. Like the Anne Rice vampires, Alucard is somewhat jealous of what differs himself from humanity, though presumably less so than, say, the Brat Prince. He refers to himself many times as a monster, the probable most noticeable of which is when he calls the Major (a man who was made a cyborg) a "man that looks like a monster" and himself a "monster that looks like a man." The point to all this? Alucard is one of those Faustian guys who realizes way after the fact what he had given up forever, and regrets it. That isn't to say he hates his current standing, as he DOES love to kill fuckers (England, bitches), but that he's a bit longing for the frail strength that humanity and mortality offer. Basically, an easier way of looking at it is; due to their mortality, humans oft have something important to fight for. Alucard, in his immortality, does not. As such, humanity's borrowed time is precious to him, and he hates it when it is taken for granted (MOST noticeably so when Anderson uses Helena's Nail to gain his own immortality, and Alucard quite clearly admonishes him for relinquishing his humanity just to be on-par with himself for their final battle.).
tl;dr Alucard loves and misses humanity. That's the long-and-short of it. Sorry for the text wall.
And yeah, Al's pretty open about the sexual nature of battle. Though, for him being what he is, it's a natural question for him to ask, since his sexual feelings and preferences mostly become beyond human comprehension(though there are exceptions like the Major and Anderson, and even they aren't quite at what his level would presumably be). It's a funny scene, though, for sure.
And chainsaw, I don't quite agree with you there. Sure, Al wasn't exactly romantic towards Seras, but I don't think he wanted to push her into what he was. Rather, I feel that (based on all the shit I explained in the spoiler tags) he admired her drive for life and Turned her partly due to that. They have a form of a master-pupil relationship going on, undoubtedly, but I never thought that Alucard had negative connotations in his motivation for taking her on as his fledgling. He's not JUST a killing machine. As for his relationship with Integra, that's explained in a similar manner. When he was first awakened by her blood, she stood her ground to him fairly well for being just a child. He serves her out of respect to her father, whom he also admired for being braver than his frail mortality would have suggested. It's not so much that he's FORCED to, since he could easily rip her apart and no one could possibly stop him, but that he CHOOSES to. Their head-butting, back-and-forth relationship is just a part of that, since Alucard always is playing some manner of game to test the willpower/strength/valor/etc. of humans.
Ratty put it much more briefly, if nobody wants to read all that:
Out of respect for the man who was able to take him on, and out of boredom from being his own master for centuries.
The simplest way to describe everything Alucard does is that he has a deep and genuine respect for humans who are able to stare death in the face and still stand strong. Whether they're afraid, or physically weak, or whatever doesn't really matter to him (though he does seem to give bonus points in some cases). All that really matters is the courage/strength of will to make that stand in spite of whatever obstacles are there.
Hey guys can you guess what D9's favorite anime/manga series is?