Does the Russian alphabet have A's and B's and backwards N's and R's because it's related to the Latin alphabet, or is that just coincidence?

short answer: Russian alphabet is not related to Latin at all. It is related to Greek via Cyrillic script. Strictly speaking, there are two entities, often mixed up - Russian alphabet and Cyrillic scripting. Cyrillic scripting was developed to write down texts in Slavic languages and it is adjusted to common features of Slavic languages (e.g. by having single letters for most common sounds).

For example, letter 'Щ' denotes sound 'shch' (requires 4 letters of Latin script).

Cyrillic scripting significantly changed over time, many letters were removed as obsolete, some new letters added.

those letters you've mentioned - 'И' - vowel sound, pronounced (in Russian) like 'i' in meek, corresponds to Greek etha. 'Я' - vowel sound, pronounced like 'Ya' - later development, wasn't originally in Cyrillic alphabet.

EDIT: meek etha

sources

Britannica - Cyrillic alphabet. Short historical note + images of letters with their approximate Latin transcription
Audio - pronunciation of russian alphabet. With pictures.
A brief history of russian language. Syllabus for a university course in linguistics. Look for "written language". Also lists books in English.