Why is Ophelia mad? Does anything she says make sense? What happens to her at the end of Act 4? What does her madness and death symbolize about the kingdom?
I believe the first commenter thoroughly addressed your initial question, so I will address the others.
Yes, Ophelia does say some things that make sense, especially in Act 4, scene 5. She begins by singing, "He is dead and gone, lady, / He is dead and gone; / At his head a grass-green turf, / At his heels a stone" (4.5.34-37). She appears to be referencing the death of her father, Polonius, and we might assume that his death, at her ex-lover's hand, is the final act that pushed her over the edge into madness. She sings another song about St. Valentine's Day which goes, in part, "Then up he rose and donned his clothes / And dupped the chamber door, / Let in the maid, that out a maid / Never departed more" (4.5.57-60). Some interpret her choice of song to reflect her relationship with Hamlet. It is never made clear precisely how far, sexually, their relationship has gone. However, Ophelia's preoccupation, in her madness, with a loss of virginity (also seen in lines 70-71) could imply that they'd slept together and therefore she would feel that much more upset by his madness and unkindness toward her. Later, Ophelia passes out flowers (real or imagined) to Claudius, Gertrude, and Laertes, and there seems to be some sense to the choice she makes for each character. The queen, it seems, receives fennel, columbines, and rue, possibly referencing her faithlessness, ingratitude, and regret. The king gets rue and daisies, possibly referring to his regret (which is "differen[t]" from that of his wife and Ophelia since he actually committed a terrible crime) and unrequited love, implying that the queen no longer loves him (or, at least, loves her first husband more). Laertes gets rosemary, for remembrance, and pansies, for thoughts; perhaps Ophelia wants to encourage him to remember their father and keep him in his thoughts.
At the end of Act 4, Ophelia does drown, but there is also evidence to suggest that she died by accident and not suicide. Gertrude explains that Ophelia was hanging flower garlands on a tree branch when "an envious sliver broke" and she fell into the water (4.7.198). Ophelia did not jump into the water of her own volition. Further, Gertrude describes her as singing old songs as she sank, as though she didn't realize her danger; she was "as one incapable of her own distress" (4.7.203). Whether or not you believe she took her life on purpose or she died by accident depends on how much you believe her to be aware of things outside her own mind at this point. Ophelia makes no attempt to save herself, which could indicate a choice to die, but if she makes no attempt to save herself because she does not realize that she is about to die, then her death appears to be accidental.
Ophelia's madness and death seem to symbolize that the rot and corruption in the Danish court must be cut out, destroyed completely, or else innocent lives will continue to be ruined and lost as a result. Marcellus was almost right when he said that something is rotten in the state of Denmark: many things are rotten. It begins with the murder of old King Hamlet in a garden which seemed like a paradise; the serpent, his brother, took his life, and then took his wife. The marriage is corrupt and incestuous from the start, as Gertrude and Claudius are, in Biblical law, sister and brother. Hamlet turns on his mother and his girlfriend, plotting the murder of his uncle and performing the murder of Polonius; his (relatively innocent) friends are pitted against him when they are hired by his uncle -- Hamlet arranges for their deaths as well; Laertes, an honest man, behaves corruptly when he dishonorably plots against Hamlet; Claudius's poison, meant for Hamlet, actually kills his own wife, and so forth. No one seems blameless, except, perhaps Ophelia. She is a casualty of the plots that swirl around her, and the Danish court must be cleansed and made pure again so that innocents no longer become infected, so to speak, by its corruption.
- In Hamlet, Ophelia is mentally "drawn and quartered" by men. As a result, she goes mad.
- First, her brother tells her not to fall for the spoiled antics of Hamlet. Since he's a prince, he acts on whims.
- Next, Ophelia is told by her father that she can no longer see Hamlet, whom she loves. She is instructed to return his love letters and remain in her room.
- Then, Hamlet has the "silent interview" with her as he first attempts to be mad. She is bothered by his disheveled appearance and worries for his mental state. This foreshadows her own impending insanity.
- Next, her father uses her as bait to discover the reason behind Hamlet's odd behavior.
- Then, Hamlet projects his rage regarding his mother's adultery onto her in his famous "get thee to a nunnery" diatribe. She is teetering on madness here.
- Then, she overhears Hamlet's "to be or not to be" monologue and may mistake him for condoning suicide, either his own or hers.
- Next, Hamlet kills her father. His death pushes her over the edge.
- Overall, Ophelia goes mad because she has no position in society. Men have made her a plaything, pulling her mind in every direction. No wonder she drowns herself. She cannot stand that corrupt and sexist double standards of Denmark any longer.