I don't know anything significant about Ancient Hebrew. Since there were different varieties of Hebrew and Aramaic in ancient times, I can't be sure whether information that I find in documents online applies to all varieties.
With that caveat, the title and first sentence of this question seem a bit strange to me, because my understanding is that Ancient Hebrew had phonemic (not allophonic) length distinctions for most consonants, not just the stop series. As you say in your answer, certain consonants were supposed to be non-geminatable in Biblical Hebrew: the Wikipedia article "Biblical Hebrew" says
Geminate consonants are phonemically contrastive in Biblical Hebrew. In the Secunda /w j z/ are never geminate. In the Tiberian tradition /ħ ʕ h ʔ r/ cannot be geminate; historically first /r ʔ/ degeminated, followed by /ʕ/, /h/, and finally /ħ/, as evidenced by changes in the quality of the preceding vowel.
But I haven't found any source that lists /s/ as a consonant that cannot appear as a phonological geminate in Biblical Hebrew. So the implication of the sources that I've read seems to be that singleton /s/ was phonologically contrastive with geminate /ss/ in Ancient Hebrew.
Phonologically contrastive consonant length was not written in the Hebrew alphabet. There are later traditions of marking length (or for plosives, the absence of spirantization) with diacritics, but these don't necessarily correspond to the distribution of length in earlier time periods. I'm not sure what evidence experts use to establish the existence of geminate consonants in particular words in Ancient Hebrew.
In terms of possible conditioning factors for a process of consonant lengthening, stress seems to be important. I found a web page that refers to "Pretonic Vowel Lengthening or Equivalent Consonant Gemination" and says that "pretonic gemination at times substitutes for pretonic lengthening" ("Phonemic Structure of Pre-Exilic, Tiberian and Israeli Hebrew Contrasted", History of the Ancient and Modern Hebrew Language, by David Steinberg). "Pretonic" in this context seems to refer specifically to the syllable immediately before the stressed syllable. Also, I don't know how schwa relates to the process described there.
I don't know the original position of the stress in the words that you mention, but I would recommend studying that.
/e/in particular the Koine orthography had a way to indicate vowel length (using epsilon versus eta).