Critically discuss the ways in which Shakespeare presents love in Much Ado About Nothing. What are the relationships like between Beatrice and Benedick, Claudio and Hero, and Beatrice and Hero? What are the relationships between the male characters in the play like?

Excellent questions! Shakespeare's Much Ado About Nothing is a unique play, and his intriguing depictions of affection between all of the characters give us much to think about.
First of all, let's discuss how Shakespeare presents non-romantic love. One of the best examples of this is the tender relationship between Beatrice and her cousin Hero. Hero loves Beatrice dearly, and we see this when she agrees to work with Don Pedro and Claudio in the plan to unite Beatrice with Benedick:

HERO: I will do any modest office, my lord, to help my cousin to a good husband.

While Leonato, Claudio, and Don Pedro seem more inclined to think of the plan as a joke, Hero is sincerely concerned for her cousin and wishes to see her happily married.
We see Beatrice's love for Hero in the way she speaks to and of her, with much more gentleness than she does to anyone else. We also see it in the tender way she defends and cares for Hero when Claudio accuses her of infidelity.

BEATRICE: O, on my soul, my cousin is belied!

She again shows her fiercely protective love for Hero when she asks Benedick to kill Claudio for her, to right the wrong. Her passionate speeches when he at first refuses again give evidence of her intense affection for Hero:

BEATRICE: Is 'a not approved in the height a villain, that hath slandered, scorned, dishonoured my kinswoman? O that I were a man! What? Bear her in hand until they come to take hands, and then with public accusation, uncover'd slander, unmitigated rancour—O God, that I were a man! I would eat his heart in the market place.

Let's look at a second example of non-romantic love in the play: the relationship between Benedick, Claudio, and Don Pedro. Seemingly the oldest of the three, Don Pedro acts mainly as a sort of mentor to the two young men, cautioning them and advising them in many different ways. Benedick clearly respects Don Pedro and values his opinion. Claudio relies heavily on Don Pedro's advice and counsel, especially in his pursuit of Hero. Don Pedro seems fond of both men and clearly wants what is best for them. He shows this by concocting the plan to unite Benedick with Beatrice and actively helping Claudio to woo Hero; he also supports Claudio in disowning Hero at their wedding.
Benedick and Claudio clearly have a close relationship, though their personalities are very different. Much in the same way that Beatrice is fond of Hero, Benedick is fond of Claudio and attempts to guide him, though more in jest than seriousness, by reminding him of the dangers of falling in love and getting tied to a woman. Claudio's affection for Benedick is obviously rooted in respect, for he repeatedly tries to get his honest opinion on things.

CLAUDIO: Thou thinkest I am in sport. I pray thee tell me truly how thou lik'st her.

Another example of Benedick's affection for Claudio is in his initial refusal of Beatrice's request that he kill him, even though he has only just won her heart and has promised to do anything for her.

BEATRICE: Kill Claudio.
BENEDICK: Ha! not for the wide world!

Though he later relents and allows himself to be persuaded to challenge Claudio for her, when he does meet Claudio, he only threatens and delays the actual action until Claudio should choose to come to him.
Now let's examine the two romantic examples of love: Claudio and Hero, and Benedick and Beatrice. We'll take Claudio and Hero first, as theirs is simpler to discuss. When Claudio meets Hero, he is at once besotted, admiring her beauty, grace, and purity of character.

CLAUDIO: Can the world buy such a jewel?
CLAUDIO: In mine eye she is the sweetest lady that ever I look'd on.

Hero's feelings for Claudio are somewhat ambiguous in the beginning, but by the end of the play, despite what Claudio has put her through, it is clear that she loves him. Consider her speech to him when she unmasks after having pretended to be dead:

HERO: And when I liv'd I was your other wife; (unmasks) And when you lov'd you were my other husband.

Gentle Hero forgives Claudio his mistakes, and they are happily married in the end, despite everything.
Benedick and Beatrice's love story is more complex. Both of them are dead set against marriage in general, and especially to each other, in the beginning of the play, though both recognize some good qualities in the other, as we see from Benedick's comment on Claudio's raptures about Hero's beauty.

BENEDICK: I can see yet without spectacles, and I see no such matter. There's her cousin, an she were not possess'd with a fury, exceeds her as much in beauty as the first of May doth the last of December.

When Claudio, Leonato, and Don Pedro make an attempt to convince Benedick that Beatrice is in love with him, it doesn't take him long to decide that he loves her too, giving us a hint that perhaps he was already interested in her. The same is true of Beatrice. She hears Hero and Ursula discussing how Benedick is in love with her. This shows that while Beatrice and Benedick made no attempt to show it, they had already begun to love each other, perhaps not in spite of, but because of, their constant verbal battles.
Their love for each other grows as the play goes on. In the scene after Hero's disastrous wedding, they both declare their love, and Benedick sacrifices his relationship with Claudio for Beatrice. At the last scene of the play, we are treated to a sweet scene where the pair have a final merry war of wits that culminates in a kiss.