I feel comfortable with the following 'natural' distinction: (a) when אלהים is grammatically linked with verbal forms, or other inflected parts of speech, in the plural, we have to understand it as a numerical plural noun; (b) when אלהים is grammatically linked with verbal forms, or other inflected parts of speech in the singular, we have to understand it as a non-numerical plural noun.
As regards the 'non-numerical plural' (as I termed it) the Grammar of Joüon-Muraoka list, at least, 6 kinds of them present in MT ('composition', 'extension', 'excellence/majesty', 'intensity', 'abstraction', and 'generalisation' [§ 136:a-j; compared it with Gesenius' Hebrew Grammar § 124a onward]).
So, in Gen 1:1 - for an example - when we encounter אלהים along with the other parts of speech (linked to it), in the singular, we know that there the common termination "-IM" cannot marks a numerical plural (as God is 'composed' by several 'elements') but it is a non-numerical plural, in this case, an excellence/majesty one. Interestingly, the Gesenius Grammar cited above says: "So, especially אלהים Godhead, God (to be distinguished from the numerical plural gods, Ex 12:12, &c. [...] That the language has entirely rejected the idea of numerical plurality in אלהים (whenever it denotes one God) is proven especially by its being almost invariably joined with a single attribute [...]." (ibid. §124b)
Your question is: "Is Elohim [אלהים] in Genesis 1:1 a personal proper name, a title, or a common noun?"
So, the answer is: taking into an account the context (also the global one) of Gen 1:1 we may say that אלהים there indicates the Creator God IEUE. Technically (grammatically), this term here indicates a non-numerical singular noun.
Obviously, different contexts in which this term is used in MT triggers a different word class (a personal proper name, a title, a common noun, and so on).