A lot can be, and has been, written on the subject, but I'll give you the short and sweet version.
Does nuclear chain reaction start in fuel pellets even before they being installed in reactor? -- No
There are several reasons why this is so.
The number of spontaneous fissions of $^{235}$U is minimal. The branching ratio for that mode of decay is $7 \cdot 10^{-9} \%$, which means that for every billion $^{235}$U atoms that decay, only $7$ of them do so by spontaneous fission. This does not produce enough neutrons to start a chain reaction.
The neutrons released from fission have too much energy to induce many more reactions. The probability of an atomic event is characterized by the associated cross-section. The cross-section for the relevent fissions of $^{235}$U at the fission spectrum average is 1.235 barns. This is not zero, but it isn't very large; compare this to neutrons in the 0.025 eV range where the cross-section is 584 barns.
Fresh fuel rods are not typically enriched very much. The exact enrichment varies depending on a variety of factors, but fresh fuel is typically on the order of 2-5% $^{235}$U; most of the rest of the fuel is $^{238}$U which is significantly less likely to fission due to neutrons in the average fission spectrum.
As to your confusion, yes, the fuel is sufficiently enriched to sustain a chain reaction; that is what it is designed for. It is designed, however, to be inside a reactor when that happens. Inside a reactor, there are other things that start and sustain the chain reaction. The primary of these is a moderator.
A moderator is a substance that slows the neutrons down from the fission energy of around 2 MeV to the average temperature of the moderator, around an eV or so. In all commercial reactors in the United States, this moderator is plain old water. Some reactors, though, use heavy water and others use graphite. Either way, the function is critical for (most) nuclear reactors. There are such things as fast reactors, but I'll let you research that on your own.
Also, for fuel pellets and rods are radioactive, how do we transport them? -- Very carefully.
Fresh fuel rods, as explained above, are not dangerously radioactive and can be handled without a great deal of extra caution. Fresh fuel is made into pellets that go into rods; the rods are assembled into assemblies which are transported to the power plant in transportation casks. These are often times not much more than a wooden box with packaging material. They can be loaded on the back of a semi or onto a train and shipped to the power plant.
Spent fuel rods are typically moved by machine and only then a very short distance into cooling pools. These pools provide sheilding while allowing the removal of excess heat generated by fission products. After many years in a cooling pond, many nuclear power companies have started to move spent fuel to dry cask storage containers. These containers provide the same benefits of the cooling ponds, but require less maintenance.
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.
Best Answer
There aren't many radioactive isotopes of Oxygen and Hydrogen and the ones that there are aren't very radioactive.
As dmckee notes, there is Deuterium formed from Hydrogen capturing a neutron, this produces D$_2$O, or heavy water. But Deuterium is stable and so doesn't cause radioactivity in itself. Heavy water is chemically a little toxic but not a radiation risk. You could produce Tritium when Deuterium captures another neutron, but the rates of this happening at fuel rod energies/intensities is tiny.
The most likely source of fuel pond becoming dangerously radioactive is a crack in one of the fuel rods allowing isotopes generated in the fuel to leak out.