I think that the most hard-to-explain idiomatic wording here is "fall in love with." After all, we don't say that we "fall in friendship with," "fall in sorrow with," "fall in appreciation with," "fall in envy with," or "fall in fear with" others, any more than we "fall in anger" with them.
And yet we don't seem to view love as existing entirely on a separate plateau from other emotional states, as is evident from the broad applicability of such phrases as "yield to anger," "yield to love," "yield to sorrow," "yield to kindness," "yield to envy," and "yield to fear."
The earliest confirmable Google Books matches for the idiom "fall/falls/fell/falling in love" is in the title of poem by Henry Howard, Earl of Surrey, who was executed by Henry VIII in 1547; the poem is headed "The Lover curseth the tyme when fyrst he fell in love." And Shakespeare uses the phrase at least four times in As You Like It alone. From As You Like It, act 1, scene 2:
Rosalind. From henceforth I will, Coz, and devise Sport: Let me see, what think you of falling in Love!
From act 3, scene 5:
Rosalind. He's fall'n in love with your Foulness, and she'll/Fall in love with my Anger.
Rosalind. I pray you do not do not fall in love with me,/For I am falser than Vows made in Wine;/Besides I like you not.
Phoebe. There be some women, Sylvius, had they mark'd him/In Parcels as I did, would have gone near/To fall in Love with him ; but for my part/I love him not, nor hate him not ;
As to why sixteenth-century (and perhaps earlier) English writers and speakers adopted the idiom falling in love, I speculate that then, as now, the intense rush of giddiness that a person suddenly and deeply in love experiences fully justifies equating that feeling with falling through the air—even to the point of not knowing whether the landing will be gentle and soft or abrupt and hard.
With emotions such as anger or sadness, people sometimes use the phrase "fall into X" (where X is the emotion in question). For example, a 1793 translation of Seneca's Morals has this passage:
All that we have to say in particular upon this subject ["Cautions against anger in the matter of education, converse, and other general rules of preventing it, both in ourselves and others"] lies under these two heads ; first, that we do not fall into anger ; and, secondly, that we do not transgress in it. As in the case of our bodies, we have some medicines to preserve us when we are well, and others to recover us when we are sick ; so it is one thing not to admit it, and another thing to overcome it.
This quotation (one of 3,510 matches that a Google Books search claims to find for "fall into anger") is especially interesting because it draws a parallel between anger and sickness—and in English we idiomatically say that someone "falls ill" or "fell sick." The notion that anger, like sickness, is an undesirable state that a person may fall into seems fundamentally different from the idea of "falling in love"—as if love were the fluid one is immersed in and passing through. To the limited extent that people speaking of falling into love, the emphasis is on the accidental or unexpected nature of the occurrence; far more often we refer to falling in love—to the exhilarating state of falling itself.
And yet, people have for centuries also recognized a close association between love and sickness, as we see in this extract from Barnabe Barnes, Parthenophil and Parthenophe (1593) [snippet]:
Both Loves and Graces strive to mend her./O never let me rest; but sigh and weep!/Never but weep and sigh! "Sick is my Love;/And I love-sick! Yet physic may befriend her!/But what shall my disease remove?"
Nevertheless, lovesickness seems generally to be identified as a form of sickness more strongly than as a form of love, and for this reason I would expect English speakers to be more inclined to speak of "falling into lovesickness" than of "falling in lovesickness."