I've always inferred that the actual gender of the object isn't always indicative of the grammatical gender.

This is true, and I've mentioned it here and elsewhere. Gender was originally a grammatical term that became a sort of euphemism for biological sex. There are some languages, famously of the bantu family in Africa, that have upwards of 13 genders (though today they tend to be called noun classes because of the word gender's newer meanings).

the Latin word "hasta" (spear) is feminine, instead of neuter.

Yes, but it is good to remember that not all nouns that end in -a in Latin are of the feminine grammatical gender. Latin agricola 'farmer', nauta 'sailor', and poeta are all grammatically masculine. And, not all nouns which end in -us are masculine: e.g., corpus, corpora, 'body' and opus, opera, are neuter and manus 'hand' and foetus 'fetus' are feminine. This leads to some funny attempts (by folks with little Latin and less Greek) to replicate plurals in English with words borrowed from Latin, e.g., *opii as the plural of opus, virii as the plural of virus, etc. The last mentioned is particularly funny as virus 'poison' seems to have been a non-count noun in Latin and had no plural form.

Ceci n'est pas un seing.