Venice can probably be described as an oligarchical Republic. Generally only the members of the wealthiest and oldest families had the resources to stand for elections, so while they may not have been aristocrats in the traditional sense, for all practical purposes they were. Social mobility was greater than many areas of Western Europe at the time but this was more due to the context of the time. Remember serfdom was a common feature of much of Europe at the time, so while a poor merchant or sailor could strike it rich, this was the exception rather than the rule. Citizens of the Republic had rights that included the right to vote and stand for elections for example. I know this is a brief overview, but I'll do my best to answer more questions.

How free would the average person living in the republic be compared to someone in say France for example.

Can you define what you mean by the word "free"? Do you mean social mobility, for example how John Hawkwood started as (probably) an English archer and ended up as leading European commander in Italy, or something else?