Dashes and Dots

What're are the dashes and dots above the letters of transliterated Names for? Does they have to be included to properly spell and pronounce the Names of the Gods?

They mark stress and vowel length. Vowel length has to be marked for true accuracy, stress is marked for precision.

How do I know which marks are the most accurate? There are so many different spellings. And once I find that, how do I make them on an English keyboard?

For Greek, it depends on how you are going to pronounce it. A transliteration of Greek, letter for letter, gives Sōkrátēs: long "o" and "e", stress on the "a". But the normal English pronunciation has a short, stressed "o" for which the traditional Socrates is quite good enough. Of course, a transliteration is reversible: if you see "ō" you know it represents omega and not omicron, but if you know Greek you know that already, and if you don't it doesn't matter!

In Sanskrit, Vishnu and Shiva are transcribed Viṣṇu and Śiva. The two accented "s" sounds are not the same, but English speakers (and those of many modern Indian languages) can't make the distinction, so "sh" is good enough as a transcription.

As for computers, that depends on your operating system. I use Linux, so for "ō" I press, in sequence, Compose hyphen o. In Windows, you have to hold down LeftAlt and type 0333 in the numeric keypad; things don't seem to have changed since MS-DOS in the 80s!! I presume that MS Word has something, better, but I've never used it.
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So if I were writing for the community of English-speaking nations, would it be alright to spell the Names as follows with the following letter construction?:

Zeus
Hera
Aphrodite
Poseidon
Demeter
Artemis
Apollon (Apollo)
Athene (Athena)
Hephaistos (Hephaestus)
Hermes
Dionysos (Dionysus)

etc, etc.

I'd say any of the forms you've listed will do. There's no "right way": all that matters is that people can recognise whom you're writing about.

It's the same for pronunciation. Plato said [demeter], Aristotle grew up saying [damater], modern Greeks say [dimitra], and I say [dimitə]!

The spellings you have are reflective of pretty much 'standard English' usage (where you have parenthesized names, the usual is the one you have parenthesized). Those are the ones that would be in an English dictionary, and the ones that students would be taught at school.

Yes, I am aware of that. I wrote the Names all in transliteration, and those that are not commonly read in such format, I put the modern Latin translation beside of it in parenthesis. Most of the Greek Names translate to English letter for letter, but some, such as Apollon, do not, and so I made sure to add the better known Apollo beside of it so people would understand better.

I've seen some Hellenists write the transliterated Names without the dashes or dots. For example, Apollon without the mark above he "o." Is it acceptable to write the transliterated Names in this fashion, with just English lettering and nothing more?