Why some reaction have no transition state?
There is nothing special in having an activation barrier between reactants and products although, obviously, very many reactions do. But some reactions, such as electron transfer, are observed experimentally to have no barrier which happens when the product potential crosses that of the reactant at its minimum (i.e. between the 'normal' and 'inverted' regions).  In fact  electron transfer is an interesting case as it is possible to control the barrier height by changing the redox potential. Some atom diatom reactions such as H+F2 also have very very small barriers.  Whether you want to call the crossing point a transition state when it has a zero barrier is a moot point. The behaviour is more like water going over a waterfall, so limited by the transport of the wavepacket along the reaction path, rather than sometimes being reflected at the barrier and sometimes crossing.