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:
Lewis. 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 Curt F. 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 𝐾=∞, then Δ𝐺∘=−𝑅𝑇ln𝐾 would turn into Δ𝐺∘=−∞. 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.
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. 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 Curt F. 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."
Roduner & Radhakrishnan. 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. This is achieved by letting the machine operate between two temperatures, where energy is provided to the system at the high temperature and partly converted to work. This can be done only when an unavoidable remaining fraction of the energy input is degraded to lower temperature and dissipated as waste heat. It is shown here that this principle is not limited to heat engines but holds more generally for many other processes in which a subsystem of the universe is to be transformed to higher order, as in the separation by distillation or crystallisation, photosynthesis, solar cells, or self-assembly...
The simple model universe of Fig. 18, the combustion of glucose, predicted that the final state of the universe will be much warmer compared with today and consists of a higher fraction of atmospheric CO2 and H2O at the cost of O2. One might therefore ask whether the global warming that we see on our planet actually reflects that we are noticeably approaching the heat death of the universe as a consequence of combustion of fossil fuels with corresponding heat and entropy production. The answer is straightforward: the direction of the effect is correct, but it does not quantitatively reflect the approach of our planet to heat death. The solar heat incident on our atmosphere is about 5 orders of magnitude more than the total energy consumption by the human population. CO2 and water produced by combustion just increase the mismatch between incident and reflected solar energy by increasing the greenhouse effect of the atmosphere, thereby amplifying the direct effect by a considerable factor.
If we're to believe that heat death is a logical conclusion of the second law of thermodynamics, then there must be more exothermic reactions to result in an ultimately stagnant system where all available energy has been dispersed as heat and atomic components are at their lowest possible energy states — this outcome, by definition, represents an excess of exothermic reactions.