The Greeks used the gods and goddesses as a way to explain the unexplainable. Their myths were the stories explaining why something was the way it was. Like with Apollo and his lover Hyacinth, when Apollo accidentally killed Hyacinth he made him part of the earth, which is where we get the Hyacinth flower and apparently they had a lot of red/pink ones because the blood from his wound is how they got the color. They used myths to explain the seasons even, with the story of Demeter and Persephone. Yes, the Romans were influenced by the Greeks in their culture, language, and mythology, to name a few. The Romans were also influenced by the Etruscans, who were in turn influenced by the Greeks. The Greek people brought their colonies to Italy pre-dating the Roman people, which is one reason why the Greeks were looked up to as the high society. The Romans weren't even around at this point. The Roman trinity of Jupiter, Juno, and Minerva orginially came from the Etruscan trinity of Tinia, Uni, and Menrva, showing that there was more influence than just the Greeks. For just about every Greek god, there is a Roman couterpart.
To the Romans, yes the gods were real. Each river had its own deity because to them, the rivers running without any aid showed that it was alive, meaning anything that moves is considered alive. Life was completely different in their time, and to just brush it off and call them "crazy" is not right. Sure some of the things that they did may seem odd to us, but that was the way it was done.