The etymological derivation of the noun fōmes, fōmitis from the base of the verb foveo is too difficult for me to answer. So in this post, I'll focus on something else in your post that I think I might be able to clarify: how the different forms of the noun are related to each other in Latin.
A fair number of Latin nouns inflect like fōmes, with a nominative singular form ending in -es /es/ and a genitive singular form ending in -itis /itis/. According to the regular patterns of Latin declension, the occurrence of -it- in the genitive singular allows you to infer the occurrence of -it- in all other forms of the word aside from the nominative singular: this includes both plural forms such as fōmitēs and singular forms such as fōmitem and fōmitī.
You can analyze fōmes as a noun with a stem ending in |et|
Even though the nominative singular form fōmes has no [t] in its "surface realization", we can analyze fōmes as having an "underlying form" built on a stem ending in |et|. (In this post, I will use bars, as with |et|, to represent "underlying forms" that may contain sounds or sequences of sounds that never show up in pronunciation, versus slashes, as with /foːmes/, to represent sequences of sounds that would be audibly pronounced.)
a rule forbidding /ts/ gives fōmes from |foːmet-s|
Starting from the stem |foːmet|, we would get |foːmet-s| by adding the nominative singular suffix |s|. But Latin does not allow the sequence [ts] to occur (except for across a word boundary, or within a few compound words such as etsi that historically contained a word boundary). In contexts where the addition of a suffix would create the forbidden sequence /ts/ or /ds/, Latin regularly replaces the sequence with either the geminate /ss/, or with a single /s/ in contexts where a geminate /ss/ would also be forbidden. Since geminate /ss/ is not allowed at the end of a word in post-Plautine Latin, |foːmet-s| is simplified to /foːmes/.
a rule reducing |e| to /i/ gives fōmitēs from |foːmet-eːs|
The oblique forms of the word, such as fōmitēs, can be derived from the stem |foːmet| by a rule of vowel reduction in non-initial open syllables. Before a vowel-initial suffix such as the nominative plural ending |-eːs|, the final |t| of the stem is syllabified into a different syllable from the preceding short vowel |e|. When the short vowel |e| occurs at the end of a syllable that is not the first or last syllable of a word, it often changes into the short vowel /i/.
There are many other examples of words that show this kind of alternation between /e/ in a syllable with a final consonant and /i/ in a syllable without one. This is covered in "Aspects of the phonology and morphology of Classical Latin" (2016), by András Cser, section 184.108.40.206. (pages 90-92).
However, the reduction of |e| to /i/ does have some exceptions: hebes, hebetis; impes, impetis; indiges, indigetis; interpres, interpretis; perpes, perpetis; praepes, praepetis; seges, segetis; teges, tegetis; teres, teretis. The -es, -etis words are greatly outnumbered by -es, -itis words that do show the /e/~/i/ alternation.
Some or all of the words in this list with exceptional genitive singular forms in /-etis/ could possibly be explained by adding further conditions to the rule of vowel reduction. Cser lists seges, teges, hebes, perpes and provides the generalization that this pattern is found in words "with stem shapes of the form C[e]C(C)[et]-"; he also indicates that the fact that perpes is a prefixed word is relevant (92).