What influences did Judaism, Christianity, the Renaissance, and the Reformation have on the rise of democratic ideas?
First, one should note that democracy evolved in ancient Athens which was neither Christian nor Jewish. Similarly, the Roman Republic was pagan as well. Neither Judaism nor Christianity affected the development of democratic forms of government in antiquity. One could even argue that by the fourth century, when Christianity became the religion of the Roman Empire, the Christian system of bishops and ecclesiastical hierarchy served to enforce imperial power and was thus, by its very nature, anti-democratic. Judaism did serve as a locus of rebellion against Rome, but that was not a democratic uprising; rather, it was both religious and nationalistic.
The key point at which some forms of Christianity became affiliated with democratic ideals was with the rise of Protestantism in the Renaissance. A major ideal of Protestantism was the notion of sola scriptura (that only Scripture and not the church hierarchy was required for salvation) and the notion of the "priesthood of all believers." Both of these ideals tended to accompany more democratic forms of church governance, such as councils of elders. The notion of all white men (but not women, and often not blacks of any gender) was implicit in the notion of all men being formed in imitation of God and thus being equal in brotherhood in Christ, although that ideal only existed briefly in the primitive church and only became part of Christian church governance in limited denominations.
Also, the Renaissance was a period emphasizing a return to classical ideals, including admiration of the Roman (pagan) Republic.
One central ideal common to all of the above is that of personal responsibility to the society one lives in. Personal duty to your society, that duty being to act in ways which positively impact your society, goes hand in hand with the concepts of personal liberty.
Judaism emphasized the obligation of moral behavior toward your neighbors, and Christianity added more emphasis on personal freeedom and responsibility toward individuals as well as society as a whole. The Rennaissance saw the rise of political philosophies again emphasizing duty to one's society and culture, and individual morality. The Reformation brought even more emphasis on the individual's personal freedom and responsibilities as the two sides of a societal coin.
Think of the priveleges you have as an independant, free person in a free society. The other side of that coin is the responsibility to use that freedom and those priveleges for the good of your neighbor as well as yourself. That is the democratic principle in a nutshell, and also the central thrust of the philosphies that have come down to us through the four influences you mention.