Taking the easier part of the question first, a copy of the Russian original version of the short story can be found here. From this text it can be seen that the word used by the author, Garshin, was "трофей" (pronounced as "trofey") which is cognate to, and has the same meaning as, the English word "trophy".
The first part of the question, concerning the interpretation of the poem, is more difficult to answer. A poem does not have a single fixed meaning, and indeed, we must all find our own meaning in a work of poetry. Your interpretation of the red flower as a symbol of love is perfectly fine, but I do not think many readers would share it.
Garshin tells us why the madman found the red flower so disturbing:
the red petals had attracted his attention... In this bright red
flower was concentrated all evil... The flower, as he saw it, ruled
over evil; it absorbed in itself all innocently-shed blood (that is
why it was so red), all tears and all the gall of humanity. It was an
awful and mysterious being, the antithesis of God, an Ahriman
presenting a most unassuming and innocent appearance.
To the madman it seemed that the flower was evil ("ahriman" is a term from Zoroasterism signifying absolute evil), and he set out to fight against it. He succeeded in destroying the first two flowers, but died after picking the third, which had grown after the first two had been picked. Rather than being connected to love, I would rather interpret the three flowers as representing a Trinity, but an evil Trinity in place of the Father, Son, and Holy Ghost.
Instead of connecting the word "trophy" to "trophy wife", I think it refers to the original meaning of a trophy:
Etymology. From Middle French trophée, from Latin trophaeum (“a sign
of victory, a monument”), tropaeum, from Ancient Greek τρόπαιον
(trópaion, “monument of an enemy's defeat”), from neuter of τροπαῖος
(tropaîos, “of defeat”), from τροπή (tropḗ, “a rout, a turning of an
enemy”).
In this case the dead flower is a sign of the man's victory over the forces of evil; a bitter struggle that took place within his own mind that ended in his death (although he died triumphant).