There's no way to really measure happiness prior to modern surveying techniques, but it does go without saying that of course there were significant political and tensions during the Italian Renaissance, as we might expect there to be in any time period. The whole concept of Renaissance gets difficult to work with if you think about it too much.

The Italian Renaissance was undoubtably an important period that more or less coincided with a span of extended peace and prosperity in the Italian Peninsula (although that peace and prosperity is only relative to earlier periods, and will need to include more than a few exceptions the further you set the beginning and endpoints). The social, cultural, and artistic achievements of this period have been enshrined in the Western Canon in part by historians of the 19th century (Jacob Burckhardt, notably) who pretty much shape all of Western historic framing anyway, but it's also important not to discount that the people in the period itself were to some extent self-aware of the period they lived in: in admittedly a notable departure from dour Medieval Philosophy insisting that the end of the world was imminent, thinkers from the early renaissance right through to its closing years were wont to reference that they had left behind a period of "Darkness" (Petrarch) and were living in a period of Rebirth (Vasari).

Sure, we can disagree with the premise that just because a few intellectuals believed they were particularly clever at reinterpreting ancient art, we shouldn't generalize about a whole period (or rather, the 19th century historians shouldn't have generalized about the whole period) but at the end of the day it doesn't matter for the purposes of your question: some people did think they were living in a period of artistic and cultural "Rebirth." I guess we can say that for these people, they must have been happy and proud to have lived in such a period.

Was this the only period of peace and stability that saw an expanded artistic and cultural output? No, the Carolingian and Ottonian renaissances were also important periods of peace and prosperity in Europe; they were shorter, sure, but they might have been wider-reaching and even more politically and socially influential than the Italian Renaissance was (the Italian Renaissance's main achievements are generally interpreted as more artistic than social or political, although social and political changes did occur). How happy were people (getting back to your question)? It will always depend. Who? Where?

In the Renaissance, as in any historic period, where there was peace and stability people prospered, and where there was war, plague, or famine, people did not prosper. I suppose rich people were generally more happy, while the numerous poor had fewer reasons to be, although I also suppose the lack of frequent revolts or upheavals that characterize might have indicated that all sorts of people were generally content (but again, we can find exceptions here too: the Florentines violently expelled the ruling Medici dynasty and organized the Bonfire of the Vanities in 1497 - unless you mark the end of the Renaissance as occurring before then). At any rate, everywhere society was grossly unequal, although an increasing proportion of city-dwelling professionals, merchants, and artisans were admittedly growing wealthier, but this trajectory had started around the year 1000 and was not limited to the Renaissance; indeed the Renaissance was the very crest of the parabola of economic prosperity, at least as far as Italy is concerned. The physical number of large and medium sized cities (and with it, prosperous city-dwellers) had also been growing since around the year 1000, so that's something else to keep in mind.

Even in this environment where professions, trades and commerce were growing, by and large the primary economic activity continued to be agriculture: Land represented the main form of capital in the economy. Those who owned lots of land were wealthy, those who owned little land (or rented land) were not. But social tensions, such as they exited, had actually been much more serious in the preceding centuries: As city-dwellers grew in numbers and prosperity, they opposed the ancient privileges of the aristocracy. However, in the lead-up to the Renaissance, the nominally inclusive institutions that emerged out of the aforementioned tensions were subverted by individual political dynasties, effectively creating the dozens of monarchies in Italy (which nonetheless remained centered on cities and their institutions). So the Renaissance represents an "end" of sorts to the "Communal" or collaborative parabola in Italy. Were people happy about this? Probably not - unless they were one of the winning dynasties, or artists patronized by these dynasties.

A final question for you to consider is the interpretation of, "Were people happy as depicted?" Depicted by whom? Happy how? Again, the Renaissance was a period of prosperity, more or less of widespread peace, and pioneering cultural and artistic output. Generally, I suppose people might have been happier than in periods where there was no peace, no prosperity, and cultural and artistic output was less significant. But ultimately, how could we say for sure?