Q:

The length of the final vowel in first declension nouns (Greek)

How can you tell whether a first declension noun ends in a short or long vowel?
Background
When the word is written and accented, I may be able to tell. (Not always.  E.g. θύρᾱ if without the macron).
But in trying to recall one, I may have all the letters, e.g. μαχαιρα, but not know the length of the final α.
Below, some things that do not seem to help.  For example, words referring to an object can go either way.

μάχαιρα (sword) v. θύρᾱ (door) - object
ἀλήθεια (truth) v. ἀπορίᾱ (difficulty) - concept
βασίλεια (queen) v. θεᾱ́ (goddess) - job title
βασίλεια (queen) v. βασιλείᾱ (kingdom) - ends in εια
ἀσέβεια (ungodliness) v. ἡσυχίᾱ (peace) - related to adjective (ἀσεβής and ἥσυχος)
ὁμόνοια (unanimity) v. οἰκίᾱ (household) - related to other nouns (νοῦς and οἶκος)
δόξα (opinion) v. ἐκδοχή (receiving) - related to verbs (δοκέω and ἐκδέχομαι)

My vocabulary is too limited as yet for any patterns to emerge should they exist.  Knowing them ahead of time would help me to sort new words I believe.

A:

There doesn't seem to be any single rule that can be used to determine without fail if a first-declension feminine noun ends in ᾰ or ᾱ just from the unaccented spelling of the nominative singular. (I would recommend learning the accented spelling of words, though, since it seems like something that you should know for other reasons.)There are apparently certain rules that apply in certain situations, and certain rules that apply in general but have exceptions.
If you look at the genitive, there is a fairly straightforward rule that can sometimes be applied.
According to From Alpha to Omega: A Beginning Course in Classical Greek, fourth edition, by Anne H. Groton:

If a noun’s genitive singular ending has an eta, but its nominative singular ending has an alpha (e.g., θάλαττα, -ης), you can be sure that the α in the nominative singular ending is short. (p. 27)

This allows you to deduce that δόξα ends in a short vowel, because its genitive singular is δόξης.
But it doesn't help with any of the other words you mention. First declension nouns whose genitive singular ends in -ᾱς can have short or long alpha in the singular nominative.
(Another thing that seems related to this is that "δόξα" is the only one of these words where a letter other than ρ, ι or ε comes before the α. In general, Proto-Greek long /aː/ became η in Attic except for after these three letters. There are exceptions to the usual distribution of -ᾱ vs. -η according to the ρ/ι/ε rule, but I don't know exactly how many or if there are any rules about which words can be exceptions.)
Greek Grammar, First Edition (by H. W. Smyth) hosted online by the Christian Classics Ethereal Library, lists the following rules and tendencies, among others, for the forms of first-declension nouns:

d. When the genitive singular has -ης, final α of the nominative singular is always short; when the genitive singular has -ᾱς, the final α is generally long.
Feminines fall into two classes:
  218. (I) Feminines with ᾱ or η in all the cases of the singular. [...]
  219.  (II) Feminines with ᾰ in the nominative, accusative, and vocative singular. The quantity of the vowel is generally shown by the accent (163, 164).
  In this class are included:
  1. Substantives having σ (ξ, ψ, ττ, or σς), ζ, λλ, or αιν before the final α show ᾰ in nom., accus., and voc. sing., and η in gen. and dat. sing. Thus, μοῦσα muse, μούσης, μούσῃ, ἅμαξα wagon, τράπεζα table, γλῶττα tongue, ῥίζα root, ἅμιλλα contest, λέαινα lioness. Others are τόλμα daring, δίαιτα mode of life, ἄκανθα thorn, μυῖα fly.
  2. Substantives in ᾰ in nom., accus., and voc. sing., and ᾱ in gen. and dat. sing.
  a. Substantives in -εια and -τρια denoting females, as βασίλεια queen (but βασιλείᾱ kingdom), ψάλτρια female harper; so the fem. of adj. in -υς, as γλυκύς, γλυκεῖα sweet.
  b. Abstracts in -εια and -οια from adjectives in -ης and -οος, as ἀλήθεια truth (from ἀληθής true), εὔνοια good will (from εὔνους, εὔνοος kind, 290).
  c. Most substantives in -ρα after a diphthong or ῡ, as μοῖρα fate, γέφῡρα bridge.
  220.  Exceptions to 219, 1:  κόρση temple (later κόρρη), ἕρση dew; to 2 b:  in Attic poetry, ἀληθείᾱ, εὐνοίᾱ, ἀγνοίᾱ ignorance, which owe their ᾱ to the influence of the genitive and dative ἀληθείᾱς, ἀληθείᾳ, etc.
  221.  Most, if not all, of the substantives in ᾰ are formed by the addition of the suffix ι ̯α or ια (20); thus, γλῶττα from γλωχ-ι ̯α (cp. γλωχῖν-ες points), γέφῡρα from γεφυρ-ι ̯α, δότειρα giver from δοτερ-ι ̯α (and so φέρουσα bearing from φεροντι ̯α), μοῖρα from μορ-ι ̯α, ψάλτρ-ια.

