Linguist here. The answer is "unlikely." I'll give some of the basic details about accents without being too technical, but I don't feel qualified to go into specifics about the evolution of the dialects you're asking about.
One's accent comes from where in your mouth you produce a sound, so (IIRC) Russians produce a /d/ with the tip of the tongue on their teeth while Americans produce a /d/ with the tip of the tongue just on the gums. Sum up dozens of slight differences like this and you have an accent.
Vowels are especially malleable and move around in our mouths constantly. For example, an American /o/ is produced about the middle of the upper mouth, about where your /g/ and /k/ are formed. Many European languages will produce the /o/ further back closer to the throat. Same with /u/. In English, the "a" sound goes through cycles of being purely "ah" and being purely "aa". For me, the difference is "father" and "hat," but that may not be true for your dialect. Anyway, try substituting the "aa" sound into "father" and you'll have something akin to the Old/ Middle English "father," depending on the time period and place.
For some modern day, actively changing dialects, go check out the Northern Cities Vowel Shift, also known as the Great Lakes Vowel Shift. This is a language naturally changing all its own.
To complicate the issue for a place like, say, Romania, they had to deal with war and invaders since the dissolution of the Roman Empire, so the incoming language speakers likely affected their pronunciation of their own language. For an example you may be able to relate to, consider the French influence on English after 1066:
English pronunciation owes a lot to French as well. Whereas Old English had the unvoiced fricative sounds [f], [s], [θ] (as in thin), and [∫] (shin), French influence helped to distinguish their voiced counterparts [v], [z], [ð] (the), and [ʒ] (mirage), and also contributed the diphthong [ɔy] (boy).
I might also add, the way we know how ancient speakers sounded is based entirely on their writing. Consider the Southern American homonym of "pin" and "pen," so a child might confuse the spelling. The same thing happens in the ancient world. If two words sound the same but are spelt differently in different places, it tells us that they weren't sure which letter to use based on what they heard. This doesn't mean that the sound had a one-to-one correspondence with the letters, only that they would approximate the letters to the sounds.
Probably not because the sounds of language change over time. For example, the uvular R was not found in Gaul but is now the standard French R.
accents are a result of language being stored in a different part of the brain. This is why you won't get one if you grow up bilingual, but will get one if you learn a language after the age of 12-14. The second language will be stored further up in the cortex, passing through the native speech filter (not being exact Here, giving an easy to understand example) that results in a skewed pronunciation you can identify because what controls the pronounciation is the original speech patterns.
Latin changed many of the European languages. Especially the "romance" languages of French and Spanish (which is why they sound similar). The Germanic languages existed BEFORE rome (all original western European languages were based on proto germanic, English is as well which is why we have many words in common with the Germans).
German is a very guttaral language, with harsh sounds. The accents in Rome would have been loosely simar to how a German sounds speaking french.
My father actually was a German married to a French woman, it provided many laughs because he did not have those smoother rolls and was much more abrupt in his patterns.