Biweekly is, in your terms, more likely to be a "noun of the adjective".
Generally, creating new words from other words is called derivation. Creating a new word by changing the word class of a word without any change in spelling specifically is called conversion. Creating a new word by attaching something to another word is called (derivational) affixation.
The two most likely orders of derivation are:
(a) week (n.) -> weekly (adj./adv.) -> biweekly (adj./adv.) -> biweekly (n.)
(b) week (n.) -> weekly (adj./adv.) -> weekly (n.) -> biweekly (n.)
Determining the order of derivation can be somewhat tricky, and giving a definitive answer is often impossible. However, it is generally assumed that more semantically complex lexemes are derived from the more simple ones.
Thus, the noun weekly ('something that appears weekly') was likely derived from the adjective/adverb weekly ('once a week'), as the former contains the latter in its meaning. Also, the result of attaching the suffix -ly is standardly an adjective or adverb, so going from week as a noun straight to weekly as a noun seems highly unlikely.
Therefore, biweekly is either the result of (a) conversion from biweekly (adj./adv.) or (b) affixation of bi- to weekly (n.). However, it's almost impossible to know which one it is.
Source: BA degree in English language & linguistics
There is no specific term for nouns that are also used as adjectives, or the other way around. The process by which one word is derived from the other, however, is called conversion, as mentioned above.
As pointed out in the comments to your question, either I propose a biweekly newsletter or I propose a newsletter to be published biweekly are fine. To avoid any confusion whether biweekly means twice a week or every two weeks, I would however suggest being maximally clear and saying I propose a newsletter to be published twice a week.