Does the Nuclear reaction happen inside the fuel rods?

You are missing the concept of critical mass. In a small amount of uranium, like a single pellet, some fraction of the atoms will decay spontaneously every second. When enough of this is put in proximity, then the emissions of some of the decaying atoms kick other atoms just right so that they fission too. As a result, more atoms fission than you would predict from just the probability of a single atom doing so.

The main difference between a small pile of uranium and a large one is that in a large one you get a chain reaction. Power plant reactors rely on this chain reaction mechanism to get the large amounts of output power. The control rods control how much the emissions of fissioning atoms can hit other atoms, thereby controlling the overall reaction rate.

In reality, this description is over simplified. There can also be moderators envolved that sortof convert some of the fission results into stuff that can kick other atoms to fission when without the moderator they would not. However, that is a aside to this question.

Uranium pellets are not dangerous if they haven't been inside of a working reactor (I wouldn't advise using them in a school chemistry class, though). During fission, there starts a buildup of various highly radioactive products (Strontium, Iodine, Caesium) and also transmutation of Uranium-238 (always present, even in enriched uranium) into various isotopes of Plutonium.

Now, after some time within the reactor, pellets are not safe to take into your hand.

The remaining question of yours: how does a chain reaction start in the reactor? A neutron source is used to create enough neutrons to initiate fission, and control rods are withdrawn in a specific order to enable rapid multiplication of neutrons until the reactor reaches a given power level. That's the basics, there are many books if you decide to be a reactor operator.

To your specific question. The nuclear fissions take place in the Uranium Dioxide fuel pellets that are stacked inside the sealed fuel rods. The geometry of the fuel rods and the moderator facilitate conversion of direct fission (fast) neutrons to (slower) thermal neutrons. 'Thermal neutrons' are much more likely to cause subsequent fissions to sustain the chain reaction. Fast neutrons become thermal neutrons by passing through the moderator before (maybe) hitting a fissionable nucleus. So any given neutron that causes a subsequent fission will, in all probability, do it in a fuel rod other than where it started. It is while moving outside of the fuel rod that the neutron is moderated or captured (by a control rod for example) providing a means to control output. So the neutrons are 'all over the reactor' but the nuclear reaction of interest occurs inside the sealed fuel tubes, in the fuel. As to handling UO2 fuel pellets or dropping one on the floor... Watch Silkwood.