After some reading and research, I can't help but notice many similarities between Old Latin and what would become the Romance Languages. For example, the case endings for the nominative and accusative in the 2nd declension in Old Latin (bonos, bonom) resemble the endings in most Romance languages moreso than Classical Latin; also the 2nd person plural possessive, vester, is rendered as voster in Old Latin, which is the same as in the Romance Languages. Finally, many Romance languages in the east like Italian, Romanian, and French (in speech at least) drop the final s; which speakers of Old Latin usually did also. Is there a relationship here at play or is this a coincidence, that the developments were separate?
Almost everything in Romance languages that comes from Old Latin passed through Classical Latin.
In the case of u/o, it's probably a coincidence that some Old Latin o corresponds to Classical Latin u and Romance o. For one thing, the sound changes from Old Latin o to Classical Latin u and Classical Latin u to Romance o occur in different contexts.
Proto-Italic short *o turned into Classical Latin u in non-initial syllables, and in initial syllables before certain consonant clusters (e.g. any cluster starting with L). There seem to have been a series of o-to-u sound changes, rather than a single immediate shift. Or at least, we can say that the spelling doesn't change all at once. E.g. the Senatus consultum de Bacchanalibus (c. 186 BC) shows u in the endings of o-stem nouns, but o in tabolam. It's usual up through the mid-1st century AD to write "o" after V/U, in words like servom and equom, while writing "u" in the endings of other o-stems, such as bonum/bonus.
Classical Latin short u of any origin turned into o in Italo-Western Romance in any syllable of a word (setting aside compliciations when certain sounds followed; e.g. outcomes vary before vowels, velar consonants, and palatal consonants).
In the Sardinian and Eastern Romance branches, short u did not merge with o; it instead merged with long u.
Examples to illustrate the differences in distribution of Old Latin and Romance o:
the participle ductus. The u in the first syllable is original, not derived from raising of Proto-Italic *o. But in Italian, it turns into o (dotto). This is not a conservation of something found in Old Latin; it's an innovative sound change. Since the same sound change fully explains the o in the second syllable, there is no reason to suppose the vowel of the ending doesn't come from Classical Latin /u/.
the noun cervus. Words ending in -vus are all o-stems and were spelled with -vos in Old Latin. In Italian, we find cervo with /o/, but in Sardinian, we apparently can find the form chervu, with /u/.
Loss of final s
The loss of final s in Old Latin is different in several respects from what we see in certain Romance languages. Word-final s in Old Latin could be dropped before a consonant, but not before a vowel, and it seems to have disappeared without lengthening or diphthongizing the preceding vowel (since short vowels before elided word-final s scan as short).
Old French final "s" seems to have been pronounced as a sibilant consonant. The loss of "s" occurred later, along with the loss of most other word-final obstruent consonants.
Italian has lost final s, but it seems to have often affected the quality of a preceding vowel.
Per Vulgar Latin, by József Herman (1967, translated by Roger Wright 2000), the loss of final s in Romania and Italy probably occurred in the second half of the first millenium AD (pages 40-41).
On the other hand, a recent paper by Béla Adamik, "The Problem of the Omission of Word-final -s as Evidenced in Latin Inscriptions" (2017), seems to conlude that there is continuity between the Old Latin s-loss and Romance languages:
The phonosyntactically determined deletion of final -s before subsequent consonants must have continuously existed all along the history of Latin and could not be suppressed irrevocably by any standardisation process. This situation might have been inherited by the Romance languages where different and complex morphological innovations led either to the discontinuation of the phenomenon of phonosyntactically determined deletion and the stabilization of word final -s (as in Western Romance), or to the completion of the deletion process and the complete loss of final -s (as in Eastern Romance).
I haven't had time to read this paper yet though.
This word seems to have a convoluted history.
The Proto-Italic form is actually reconstructed as *westero-, per Michiel de Vaan (page 691). He writes "Latin has analogical voster after noster", but then starting around 150 BC, we see the form voster get replaced with vester due to a sound change that turned /wo/ to /we/ before r, t, s in many words.
Then in the Romance languages, we see forms based on vos- again. While this conceivably could be a survival of an archaic form, I don't think there's evidence for this versus another round of analogy with vōs and noster.
one possible Romance archaism: first-declension nominative plural -ās
I've seen it argued that the Proto-Romance nominative plural ending -as (attested in Old French as the feminine cas sujet plural ending -es) may be an archaism. I don't have a good cite, but it's mentioned in this Wikipedia article: Romance plurals
That article says nominative plural -ās is attested in Old Latin, but I don't know how common it is compared to -āī (the source of Classical Latin -ae), which was formed by analogy with the second-declension nominative plural.