Japanese morphemes

Hello everyone! I've been stuck on one of my linguistics exercises for a long time now, cause basically Japenese is irrelevant to mu studies, but one of my classes actually has to do with a lot of different languages (linguistics).
Anyway, I would be so grateful if somebody actually helped me with morphemes.
The task is to divide the words into morphemes (based on the data we have). I will also provide my assumptions and I will mark them with '*' sign.

katai - 'hard' - *[kata-i]
katameru - 'harden [something]' - *[kata-me-ru]
katameta - 'hardened [something]' - *[kata-me-ta]

kiyoi - 'pure' - *[kiyo-i]
kiyomeru - 'purify [something]' - *[kiyo-me-ru]
kiyometa - 'purified [something]'- *[kiyo-me-ta]

hiroi - 'wide' - *[hiro-i]
hiromeru - 'widen [something]' - *[hiro-me-ru]
hirometa - 'widened [something]' - *[hiro-me-ta]

Can somebody tell me am I right with my assumptions? Also, based on the data, can somebody tell me if I'm right with assuming that '-ru' is a morpheme which is a nontense aspect [I mean, that it's 'an infinitive]. Then '-ta' means past time aspect, and that '-me-' is an indication of [something]? And the morpheme '-i' - is it always included in words which are adjectives?

I would be very grateful if someone actually told me whether I'm right or not.

The breakdown into morphemes is correct, though I'm not too sure about "-me-ru" (I mean the breakdown into these individual words.)

Verb form ending in "-ru" is called "dictionary form," as it is the form found in dictionaries, so in a sense it is an infinitive. However, you cannot really say so as it also serves as the present/future aspect.
And yes "-ta" is the past aspect.
"-i" ending is for (one of the two types of) adjectives. But actually this "-i" conjugates, interestingly.

"-meru" means "to make (something) (into certain condition)," but cannot be used with all adjectives consistently. And you cannot really break down "me" + "ru" and say that "me" (by itself) has that meaning.

AK: And you cannot really break down "me" + "ru" and say that "me" (by itself) has that meaning.

I agree.

If you have learned that when a Japanese simple word or suffix has its stem and the rest they are always morphemes, logically "~meru" in this dictionary form is divided, as in your assumption, into "~" + "me" + "ru." (This way is adopted in kana-kanji conversion software for a computer to analyze Japanese words.)

However, my understanding is that a morpheme refers to a fundamental semantic unit, i.e. a unit which does not include a smaller element which can convey a certain meaning.
I feel that an adjective "katai" or a suffix "meru" in this dictionary form is not divided, because here I do not see a certain meaning in "i" or "ru."
I take S "katameta" something as S did "katameru" something, not as S did "katame" something. I consider that here a morpheme "ta" is added to the stem of "katameru," not replacing a morpheme "ru."

We have another "me," which works as a morpheme meaning {relatively} when added to a stem of an adjective.
When we say "katame" it sometimes may be a noun for a verb "katameru" but in many cases it is a combination of the stem of "katai" and the morpheme "me." For example, "tamago o katameni yuderu" means {to boil egg relatively hard}.
On the other hand, when we say "kiyome" it is always a noun for a verb "kiyomeru."

By the way, we use "kiyomeru" and "kiyome" usually for a cultural meaning. For example, "kiyome no gishiki" means {purification rite}. We do not say "te o kiyomeru" if the main purpose is to prevent us from getting infected with certain viruses, in which case we may say "te o arau" {to wash hands} or "te o shoudokusuru" {to sanitize hands}.

omotenashi,

There are linguistic discussions on this topic, but there are linguists who classify the "-i" ending of "i-adjectives" as a morphone indicating/marking that the word is an adjective, while the part that comes before the "-i" ("taka" in this case) conveys the meaning of "high." If you think about it, "i-adjectives" ("keiyoushi" in Japanese, as against "keiyou doushi") conjugate themselves, as "taka-i", "taka-kunai," "taka-katta," and "taka-kunakatta."

But actually I am not that fully familiar with the different schools of thought here to discuss this topic more, or to compare, for example, Japanese grammar vs. English grammar.