The reconstructed pronunciation corresponds to the Classical Greek of Athens, which may have not been exactly the same as Homer's original pronunciation, if that's what you're interested in. About the vowel, it might be helpful to say that in terms of phonemes, Classical Greek had five short vowels and seven long vowels:
/a/, /a:/ <α>
/i/, /i:/ <ι>
/y/, /y:/ <υ>
/e/ <ε>, /e:/ <ει>, /ε:/ <η>
/o/ <ο>, /o:/ <ου>, /ɔ:/ <ω>
So like you said α, ι, υ can be either short or long, but ε and o are always short, while η and ω are always long.
ThomasNoronha wrote:4. Η and Ε are the ones giving me more trouble. Some references state they are the same with length differences, which I've already discarded. Others say Η represents an open "e" sound (IPA: /ε/ like "pet" (en) or the second "e" in "élève"(fr) or the Portuguese and Spanish "é") and that Ε stands for a somewhat more closed "e" sound (IPA: /e/ like the French "beauté", the German "Seele" or the Italian "stelle").
It's clearer for the long vowels in that ει is a close e and η is an open e. As for ε, the evidence suggests a more close pronunciation, e.g., ε + ε contracts to ει. But many η originally come from long a, so some suggest that in earlier times (perhaps for Homer) it was /æ:/.
However, others will state the exact opposite: that Η stands for the closed /e/ sound and that Ε stands for the open /ε/. I got really confused there.
Part of the confusion is that the sounds changed over the course of Ancient Greek, so η was definitely /e:/ at a later time on its way to becoming /i/, which is its modern day value.
Further, some sources state that Η stands for IPA: /ə/ like in arise or in Spanish mañana or Portuguese amanhã.
This I've never seen. I wonder what sort of evidence there might be for this.
5. About Ω and Ο I also ended up confused about what letter stands for which sound. We've got the closed sound (IPA: /o/) and the open one (IPA: /ɔ/). Classic texts say that Ο should go as obey and Ω as bone, which ads to my confusion because I read those vowels exactly the same way (though I'm not a native speaker).
Those words have the same vowel for me as well, which is why I always when books use "as in" to describe pronunciations. But the situation is similar to the e-vowels. ω and ου are long vowels with ω being open and ου being close (originally /o:/ but later /u:/), while ο was just short, but the evidence is that it was close.
6. Concerning ΕΙ, apparently it is not a diphthong but a digraph for a monophthong. That's all very well, but what is the resulting monophthong?
7. ΟΥ is clearly a monophthong, but ΩΥ is still a diphthong right? I suppose I'll have to wait for an answer on 5. to understand this one.
Well, in Homer's time, there was both the monophthong /e:/ and the diphthong /ei/, which later merged and that's why they're both written ει, similarly with ου representing both original /o:/ and /ou/. I don't know if it's possible in every case to know what ει/ου represent, so I wouldn't recommend trying to distinguish them, but it might be good to know.
ωυ was a diphthong, yes. It's similar to how ει became a monophthong but ηι / ῃ remained a diphthong (until it later became the same as η).
What is the sound difference between Ρ and ΡΡ? I have no problems pronouncing ΡΩ but I'm not sure what to do with two of them together.
I've seen both the suggestion that it was tap vs. trill like in Spanish or short trill vs. long trill as in Italian. Personally I can't consistently tell the difference between those, but as long as ρρ is longer, you'll be accurate.
9. What are the phonetic values of Σ and in which positions inside a word are they used? In Portuguese it's a pretty messy stuff, as depending on position and etymology a sibilant like in "stop" can be graphed: "s", "ss", "ç" or "c". Also at the end of words an "s" will have a more ush-ush sound like "should".
The only thing with σ is that it is voiced before voiced consonants.
10. I read some article on the letter Ν and ended up confused. Has it always the same value? For example, at the opining of the Iliad, how do you pronounce the second syllable of ΜΗΝΙΝ (μηνιν), specifically what do you do to the second N. In Portuguese we nasalate vowels so I always tend to do this. Another example, Iliad 4: ΗΡΩΩΝ (ηρωων); the final ΩΝ, should it be read like "on" (english) or "on" (french)?
11. I understand that some N turned into M when before certain letters ore as a result of juxtaposition of NM. Nevertheless, they're to be read as graphed, that is, as M, right?
There's no evidence of nasalization that I know of, and final ν has been preserved as /n/ in many cases even to modern times. So both ν in μηνιν would be the same. Now ν does assimilate like you said it becomes /m/ in front of labial consonants and /ŋ/ in front of velar consonants and that probably happened even across word boundaries. There's also evidence that at least some point it also assimilated to λ and ρ.
12. On doubled consonants, are we to prolong the value of the consonant or that of the vowel before it? In either case the idea of a double consonant is to make the first syllable heavy/long, isn't it?
The consonant should be prolonged, not the vowel and yes, a double consonant always makes the preceding syllable heavy.
13. As to the FΑΥ/ΔΙΓΑΜΜΑ is it never represented in the editions of Homer works? And if not why exactly? It seems it was read at the time, it only lost it's place in Classical Greek. Is it always replaced by a spiritus asper like in FΑΝΑΞ/ἀναξ?
The sound just didn't exist any longer when the works were written down and the letter had fallen out of use (it was preserved for other dialects where the sound persisted). There are many traces of it in Homer but I don't know when it dropped out of pronunciation. I think you meant lenis rather than asper, which is the usual result of /w/ at the start of words -- the rough breathing usually goes back to /sw/. Within words it has various results, like κουρη for original κορwη, which became κορη in Attic.
14 The same question mutatis mutandis regarding the letter QΟΠΠΑ.
The situation here is different, since koppa was just orthographic. Like many languages Greek evidently had a phonetic difference in the /k/ before back vowels and before front vowels, so it was natural to adopt both koppa and kappa. But the difference wasn't phonemic so the practice of using koppa didn't persist and kappa alone was used.