χαῖ?ε ?ν τῇ ἀγο?ᾷ φιλίαι
I am having trouble memorizing nouns as either long or short-vowel 1st declension, and would like to know which remedy is most advisable. Should one memorize both the nominative and gentivie forms of nouns, so that one can triangulate, or should one memorize it by basically associating the phrase “1st declension long” or “short” with it, as appropriate?
Thanks in advance, this is somewhat of a thorn in my posterior. 
Well, I don’t see any way to do it without actually memorizing both the vowel length and the genitive. Consider these three 1st declension nouns:
χώ?α
ὑγίεια
θάλαττα
All three decline differently. If you remember only the vowel length and not the genitive, you will miss differences between the latter two, and if you remember only the genitive and not the vowel length, you will miss differences between the first two.
χώ?ᾱ, χώ?ᾱς, χώ?ᾳ, χώ?ᾱν
ὑγίειᾰ, ὑγιείᾱς, ὑγιείᾳ, ὑγίειᾰν
θάλαττᾰ, θαλάττης, θαλάττῃ, θάλαττᾰν
One helpful thing I’ve learned is is to use what we know about the history of Greek. For example, you’ll only have a long α after a vowel or ?, since after any other sound it would have become η. So if the previous letter is another consonant, as with θάλαττα, you know that it’s short. This is also useful in that it lets you know whether the genitive and dative endings will have η or α.
Also, if you learn the accents (which I’d recommend anyway), you can use the various constraints on the accents to distinguish short and long α. For example, you can only have an accent on the third-to-last vowel if the final vowel is short so ὑγίεια must have a short α. It also lets you know that μοῖ?α has a short α and χώ?α a long α because of the circumflex vs. acute accent.
So these “rules” distinguish edonnelly’s examples on the basis of the nominative singular, and if I remember correctly when I looked into it, they work in almost all cases (especially if you assume that it’s a long α after a vowel or ? unless the accentuation says otherwise), so I think it’s pretty useful. In general, however, it’s useful to memorize the genitive with nouns because in many cases you can’t tell just from the nominative singular how a particular noun will be inflected (both in terms of what its stem is and even which declension class it belongs to).
I was taught the (so-called and by my teacher invented) RIE-rule (Rie is a girl’s name in Denmark), that all stems ending in -ε, -ι or -? kept the alpha all the way through the paradigm.
The short alphas are mostly products of the ancient suffix -ja, right? And thus must stems in -ι have a short alpha.
The same with stems in e.g. -αιν-, like μέλαινα (I know that’s not a noun, but it’s close enough)
I don’t know at all if I’m on the right track, I just think I’ve read something about that -ja suffix. Can anyone tell me whether this is gibberish?