> it usually indicates the palatalization or softening of the preceding consonant

This is another area where I think something strange is going on between the traditional terminology and the reality.

So, I read (without direct personal experience) that Spanish speakers feel strongly that there is a real difference between "ñ", the Spanish spelling of a palatal nasal stop, and "ny", which would be the spelling of an ordinary dental/alveolar nasal stop that just happens to be followed by a palatal glide.

A doubter might suspect that "ny" would be reduced to "ñ" in fluent speech. In either case, this is not a difference that an English speaker can hear.

I watched a series of "learn Russian" videos on youtube, from a native Russian, and her example of consonant softening was to draw a contrast between the English words "beauty" (with 'soft B') and "booty" (with 'hard B'). But this is much less ambiguous than the case of Spanish ñ - there is no possible way that the place of articulation of the /b/ gets drawn back to the hard palate. That would sound nothing like a /b/. It seems mandatory, to me, to analyze what is called a "soft B" as being an ordinary /b/ with following or partially coarticulated /j/. [This suggests an obvious followup question: is it possible for a Russian word to end in "вь"?]

So it seems more natural to me to say that "е" always includes a /j/ before the vowel, and that /j/ may combine with a preceding consonant to produce a palatalized consonant, if that preceding consonant allows. This analysis would be greatly strengthened if the /j/ continued to be present in words where a letter "е" was preceded by the hard sign, but I don't know enough to provide an example of that or to say that it can't happen. [Actually, that particular analysis would fail in that case, but it would strongly suggest a new analysis in which "e" is always a two-phoneme sequence /je/, and the preceding consonant is completely independent of the glide, but consonants that are already palatalized may absorb the glide, such that there is an automatic reduction of тье -> тьз.]

> but in some words, in the middle of the syllable it indicates the vowel [ɛ] and does not change the preceding consonant

Does this happen with preceding consonants that are not inherently soft? e.g. my understanding is that ш is hard while щ is soft (and the sounds are otherwise equivalent) - on this analysis, ще would involve no change to the consonant because the change is already baked in to the spelling, and ше would be more or less nonsensical.

> So how do you represent this when transcribing into the Latin alphabet in a way that more-or-less reflects both the original Russian spelling and original Russian pronunciation?

I don't think this statement of the goal can explain why it's standard to transcribe the name as "Lena". That fails to reflect the spelling, which in Russian uses "е", and it also fails to reflect the pronunciation, which is not equivalent to Лъэна. When the spelling calls for some kind of palatalization, and the pronunciation involves some kind of palatalization, how do we get a transcription that calls for no palatalization?

And as you note:

> which is both not quite accurate and not consistent with how other Russian vowels are transcribed

Why is the transcription of е so different from the transcriptions of я and ю? What goal is that serving?