Misconception. Rhyme and ρήμα are different, because they have different etymology. Ρήμα comes from ancient verb είρω, that means 1) Join pieces together putting them in a row 2) speak. Both ρήμα does. See: https://el.wiktionary.org/wiki/%CE%B5%E1%BC%B4%CF%81%CF%89 It is impressive that the root 1) ser- is the same with the word "sermon" and "series" and 2) werh₃-, *wer- (μιλώ, λέγω) is the same with the word "word", I guess, as it is shown in the above mentioned link :) See also: http://www.etymonline.com/index.php?term=sermon
Conjugation of the ancient extremely irregular verb λέγειν (infinitive), that means μιλώ, λέγω. Present: λέγω, Imperfect: έλεγον, Passed: έλεξα and είπα, Present Perfect: είρηκα, Passed Perfect: ειρήκειν. There is also a second passed tense with present suffixes and two forms (!): λέξω and ερώ. We find here all the roots you mentioned and the modern greek words λέξη (word) and ρήμα (verb, which also means word in ancient greek). Nice and complicated don't you see, like a spiral.
The strange thing is that the Ancient verb λέγω, in Modern Greek λέω=speak, sometimes is also λέγω, has abnormal future ερώ and present and perfect είρηκα and ειρήκειν. Notice that these tenses are composed analytically in Morern Greek with the use of the θα ane the verb έχω (see the tree in DL). That means that the Ancient Greeks "borrowed" these tenses from the verb είρω which root comes from the PIE language common root mentioned above. Greek language helps to understand the roots of many words in other languages of Latin, German or Slavic origin :)
it seems that greek ρίμα comes from italian rima, which means rhyme and comes from the latin rhythmus, which is actually the greek ρυθμός that you have mentioned above. So ρίμα is actually an αντιδάνειο (= repatriated loanword), that is, an older greek word, which travelled to other languages and came back to modern greek with a new form and/or meaning (source: http://phdtheses.ekt.gr/eadd/handle/10442/23156?locale=en).
No, it says that the origin of this word is from the ancient noun "ρυθμός", coming from the ancient verb "ρέω"=flow, that is the same in Modern Greek. Take a look at this: https://el.wiktionary.org/wiki/%E1%BF%A5%CF%85%CE%B8%CE%BC%CF%8C%CF%82 . It is a different verb. The verb ρέω has a common PIE root, https://el.wiktionary.org/wiki/%E1%BF%A5%CE%AD%CF%89 , that is connected with the word "stream". Different :) At least I think so.
YES! You are all right. Ρίμα is borrowed from Italian which took it from Greek ρυθμός. Details:
rhythm (En), rytm (Sw) from rythmus (La) from ρυθμός (Gr) from ρέω (to flow, stream) + θμος
ρέω from ρέFω (ancient Gr) from srew (I.E.) from srew we also
have stream (En), Strom (Ge), ström (Sw) from
rhyme (En), rimã (Ro), rim (Sw) from rime (Fr) from rithmus (La) from ρυθμός (Gr)... go to the first row
ριμα (Gr) from rima (It) from rithmus (La) from ρυθμός (Gr)
sources: Babinióti's dictionary and
Online Etymology Dictionary http://www.etymonline.com/index.php?allowed_in_frame=0=rhyme