Learn English – Is logicalness a valid word and if not what word can be used instead

No, there's no "official" source of English vocabulary. Generally, you're allowed to form adverbs by adding -ly to adjectives, so your first guess should be that mediocrely meaning "in a mediocre manner" is a word. Then check to see whether people use it like this from Larry McMurtry and the West: An Ambivalent Relationship by Mark Busby:

In high school I did a surprising number of things mediocrely. Among these were baseball, basketball, 4-H work, tennis, track, ready writing, editorial writing, extemporaneous speech, drama, declaiming, trombone playing, and debating.

The google reports almost 40K uses online. Even allowing for instances of debate over whether it's a word, that leaves enough to validate your first guess.

Devotement is an outdated, less common variant of devotion:

• The action of devoting, or fact of being devoted; devotion, dedication.
  • 1604 Shakes. Oth. ii. iii. 322 He hath deuoted, and giuen vp himselfe to the Contemplation, marke, and deuotement of her parts and Graces.
  • 1852 Wayland Mem. Judson (1853) i 29 His own personal devotement to the missionary cause.

OED

Ngram: devotement vs devotion.

Ngram: devotement.

It origin is probably due to the practice of adding the suffix -ment to verbs to form nouns which starred from the 16th century:

-ment :

• suffix forming nouns, originally from French and representing Latin -mentum, which was added to verb stems sometimes to represent the result or product of the action. French inserts an -e- between the verbal root and the suffix (as in commenc-e-ment from commenc-er; with verbs in ir, -i- is inserted instead (as in sent-i-ment from sentir).

• Used with English verb stems from 16c. for example merriment, which also illustrates the habit of turning -y to -i- before this suffix).

(Etymonline)

Devotion (n.) has an older origin:.

• early 13c., from Old French devocion "devotion, piety," from Latin devotionem (nominative devotio), noun of action from past participle stem of devovere "dedicate by a vow, sacrifice oneself, promise solemnly,"

From which devote (verb)

• 1580s, from Latin devotus, past participle of devovere (see devotion). Second and third meanings in Johnson's Dictionary (1755) are "to addict, to give up to ill" and "to curse, to execrate; to doom to destruction."

and later (17th century) "devotement".