Answering "does every reaction have a reverse reaction?" (are there more endothermic or exothermic reactions):
You've actually hit on a topic people have been exploring for nearly a century:
<PERSON>. A New Principle of Equilibrium. PNAS, 1925, 11(3),179-183
The Law of Entire Equilibrium — Thus I am led to propose a law which in its general form is not deducible from thermodynamics, but proves to be compatible with the laws of thermodynamics in all cases where a comparison is possible. It may be called the law of entire equilibrium, and may be stated as follows. Corresponding to every individual process there is a reverse process, and in a state of equilibrium the average rate of every process is equal to the average rate of its reverse process. The rate at which one group or set of groups L goes over into another group or set of groups L' is the same as the rate at which the groups L' go over into the groups L. Moreover if there are various paths by which the first process occurs, there is an equal number of paths by which the second process occurs, and the rate is the same in both directions along every path. This will be true no matter how detailed are the specifications which define the several groups and the several paths.
The implications of this concept (and your question) have been well-elucidated by SE user <PERSON> in two posts:
Is every chemical reaction in equilibrium?
Commenters suggest that "irreversible" reactions do not have an equilibrium. This is true, but tautological. In the real world, all reactions are reversible, at least to a (perhaps vanishingly small) degree. To say otherwise would violate microscopic reversibility. A reaction that was 100% irreverible would have an equilibrium constant of infinity. But if =∞, then Δ∘=−ln would turn into Δ∘=−∞. So to get infinite energy we would just have to use 100% irreversible reactions! Hopefully the problems with the idea of "irreversible" reactions are becoming apparent.
<br/
Spontaneous/Non-Spontaneous Reactions and Reversible Reactions
* All reactions can be viewed as reversible from a mathematical standpoint, if the reacting system is "big enough". Reactions that are highly spontaneous in the forward direction mean that at equilibrium, the number of "reactant" molecules will be very very small (but not zero!).
* Many times, "big enough" would mean astronomically big, such as bigger than the Earth. 273
In these cases, reactions can be regarded as "irreversible", although there is no strict, universally agreed upon boundary between "reversible" and "irreversible".
This should implicitly answer your question, as a 'reversed' exothermic reaction is an endothermic reaction. If every reaction is fundamentally reversible (on some scale, whether or not it's reasonable, as <PERSON> describes), then there are an equal number of exothermic and endothermic reactions.
Answering "which occurs more frequently: exothermic or endothermic reactions?" (are there more endothermic or exothermic reactions?):
This is an interesting way to look at things. Considering the ever-increasing entropy of the universe and the favorable relationship between entropy and heat, I would guess that there are more exothermic reactions. Exothermic reactions represent energetically favorable transformations and are more spontaneous than endothermic reactions, so it stands to reason that there are "more exothermic reactions."
<PERSON> & <PERSON>. In command of non-equilibrium. Chem. Soc. Rev., 2016, 45,2768-2784
2.7 Heat death: the ultimate end of an evolving universe
From the second law of thermodynamics we know that the entropy of the universe will increase for all spontaneous processes. Since the total possible reversible heat flow is limited by everything having the same temperature, and when no chemical or nuclear reactions are available anymore which can produce further heat, entropy will reach a maximum but will not go to infinity. This extrapolated end of a universe that evolves further in time has been dubbed heat death. But how will this final state look like? This is a question that is complex to answer. Considering an extremely simplified model universe helps to get a rough idea.
We first consider a closed system consisting of 1 mol of glucose and 6 mol of molecular oxygen at 25°C and 1 bar pressure (see Fig. 18a). The equilibrium state at the same temperature and pressure consists of the system after complete combustion to 6 mol of CO2 and 6 mol of liquid H2O. The heat of reaction of 2805 kJ will be dissipated from the system...
5 Concluding remarks
Heat engines produce work that is typically used to drive processes which do not occur spontaneously. 273