The death of Socrates by Jacques-Louis David, 1787
The oft-heard claim that "our morality comes from God" or "without God, people would act out of control" is one of the oldest and most heinous lies used to scare people into giving their hard earned cash to the clergy. The right wing propaganda website conservapedia (which I won't link to) gives us this gem: "Not possessing a religious basis for morality, atheists are fundamentally incapable of having a coherent system of morality". Anyone who parrots these claims is a bald faced liar or flaming idiot, and you shouldn't be ashamed to call them out.
Socrates, the ancient Greek philosopher and social critic famously asked religious expert Euthyphro, "is the pious loved by gods because it is pious, or is it pious because it is loved by gods"? Translation into modern language: "Is what is morally good commanded by God because it is morally good, or is it morally good because it is commanded by God"? While Socrates's heresies and social commentary (including this one) earned him the title "gadfly" from Plato and ultimately lead to his trial and death, the legacy of this question remains with us to this day.
The pagan gods of Ancient Greeks were many and so, presumably, they had different tastes. Today though, all Jews, Christians and Muslims profess to be followers of the same Abrahamic God. Still, this God seems to have an awfully hard time making his mind about anything, as his making different and often contradictory claims about what he likes or doesn't like. As Abraham Lincoln famously said regarding the question of emancipation, "I am approached with the most opposite opinions and advice, and that by religious men who are equally certain that they represent divine will. I am sure that either the one or the other class is mistaken in that belief, and perhaps in some respect both(!)...if it is probable that God would reveal his will to others...it might be supposed that he would reveal it directly to me". Truly, God could not speak with one voice on slavery. For a more thorough discussion on more recent issues on which God seems to be having an awfully hard time "having a coherent system of morality", check out this article in Psychology Today. It turns out that anytime there is some controversy, "there are people of faith on both sides of the issue" who are positive God is on their side. Things have changed very little from the days of Lincoln. For a taste of the most basic questions of morality that God gets wrong (and not just in the Old Testament, but New Testament), check out this video by Dan Savage (even though he subsequently made a partial retraction so as not to cause "offense".)
Turning to the second question, is there reason to think that without religion people would behave badly? I have found it frustrating the answers based on statistics do not seem to persuade in the least those who have already made their minds. For instance, if you point out to them that crime rates in the most godless region of the world, Scandinavia, are actually lower than the US, this is the kind of answer I once got: "Oh, but you cannot compare the masses in the US to the Scandinavians, they are held back from crime only by religion". While New Atheists are often accused of arrogance, it is their critics that, as far as I can tell, are the most patronizing and presumptuous people in this world.
But as we learn more, there is less that we need to guess on. According to a new study, generosity toward others exists in the secular people as well as religious people. The difference is not that the religious are generous and the secular aren't; rather, it is that generosity in secular people is often directed by compassion (the feeling of empathy towards others), whereas in religious people it is controlled by other factors, which may include "doctrine, a communal identity, or reputational concerns". In other words, it is not that people will behave badly if they are less religious; it is that they are more likely to actually help the poor and needy, rather than pay for maintenance of expensive churches/mosques/synagogues and salaries of the clergy.