Corrigendum:
The OP has provided a correction to equation VII, which should read:
$$ B_0(1+\sin \varphi)\left[\left(\frac{\pi}{2}+\varphi\right)-\sin^{-1}(x+y)-(x+y)\sqrt{1-(x+y)^2}\right] \\
- \left[1+\sin \varphi - (x+y)\right]\cdot T_H^4 + \frac{q}{(4\pi + 8\varphi)R^2 \sigma(1-\gamma)} = 0 \tag{VII}$$
These equations present the nonlinear system to be solved in terms that match the intended (scientific or engineering) application.
Before jumping into software development to "solve" the system, which will in essence rewrite the equations in a form suitable for programming, it is a good idea to do an intermediate rewriting of the equations in a form suitable for mathematical analysis of the conditions which imply existence of solutions and their possible uniqueness.
While the distinction between physical constants and physical parameters is crucial to the application, it will be simpler for the sake of analysis to recast the equations using fewer terms (and thus losing in all likelihood the careful distinction between them as regards the application).
Note for example that the "constant" $B_0$ only appears multiplied by a factor $(1 + \sin \varphi)$. Closer examination reveals we can divide through by the combined factor in equations V,VII,VIII in a way that reduces a good bit of "clutter".
Related observations can be made about the unknowns of the system. We already commented that $x,y$ are explicitly determined by equations XII and XIII as expressions in $T_L,T_H$. Thus those unknowns can be eliminated by substitution. Furthermore $T_L,T_I,T_H$ only appear in the equations as fourth powers, so it will be simpler just to express conditions in terms of those fourth powers directly.
Thus our first task is undertaking a mathematical rewrite of the equations in hope of simplifying the analysis of existence and uniqueness of solutions. Reasonably we can anticipate this will also facilitate the programming of solutions in a later undertaking.
For this particular system of equations we are indeed fortunate; simplifying the equations allows us to solve for the unknowns one at a time (not simultaneous as with most nonlinear systems).
To see this let's start with equation V, with only a minor simplification:
$$ B_0(1+\sin \varphi)\left[(\sin^{-1} x) + x\cdot \sqrt{1-x^2}\right]
-x\cdot T_L^4-\frac{q}{(4\pi + 8\varphi)R^2\sigma(1-\gamma)} = 0 $$
Note that the last term in the left hand side is just a constant, although it depends on the parameter $q$. Thus we can divide through by $B_0(1+\sin \varphi)$ as suggested earlier and move the resulting constant term to the other side of the equation (with a change in sign):
$$ (\sin^{-1} x) + x\cdot \sqrt{1-x^2}-x\cdot \frac{T_L^4}{B_0(1+\sin \varphi)}
= \frac{q}{B_0(1+\sin \varphi)(4\pi + 8\varphi)R^2\sigma(1-\gamma)} $$
Now the right hand side can be replaced by a single constant, although we might want to use a notation that exhibits its dependence on both parameters $q$ and $\varphi$:
$$ L(q,\varphi) := \frac{q}{B_0(1+\sin \varphi)(4\pi + 8\varphi)R^2\sigma(1-\gamma)} $$
Moreover the left hand side can be further simplified (as a Comment by the OP may have indicated) using equation XII. Squaring both sides of it and multiplying by four:
$$ \frac{T_L^4}{B_0(1+\sin \varphi)} = 4(1-x^2) $$
This looks like it was designed for the purpose of eliminating unknown $T_L$ from the simplified equation V. Substituting in the left hand side and also in the right hand side:
$$ f(x) := (\sin^{-1} x) + x\cdot \sqrt{1-x^2}-x\cdot 4(1-x^2) = L(q,\varphi) $$
Remarkably the left hand side now depends only on $x$ while the right hand side is a value easily computed from parameters $q$ and $\varphi$, $L(q,\varphi) = q/[(1+\sin \varphi)(4\pi + 8\varphi)C_L]$ where:
$$ \begin{align*} C_L &:= B_0 R^2 \sigma (1-\gamma) \\
& = (4.1\mathrm{E}9)(6.6\mathrm{E}6)^2(5.67\mathrm{E}{-8})(0.6) \\
& = 6.07583592\mathrm{E}{15} \end{align*} $$
Stepping back from the equation itself, consider the domain allowed for unknown $x$. Clearly we need $x\in [0,1]$, although not all those values admit solutions to the remaining equations.
As a check on our work so far, let's graph the function $f(x)$ on $[0,1]$.
Fig. 1 A graph of $f(x)$ on $[0,1]$
The sample values given in the Question for parameters $q,\varphi$ suggest that the values $L(q,\varphi)$ will be positive. In any case the equation $f(x) = L(q,\varphi)$ has a unique solution $x\in [0,1]$ exists when $0\lt L(q,\varphi) \le \pi/2$. For the given sample values we can find such a solution $x$, such as the entry $q = 1.0\mathrm{E}{15}$ and $\varphi = 0$, for which:
$$ L(1.0\mathrm{E}{15}, 0) \approx 0.0121319 $$
The numerical solution $x \approx 0.745165$ can be found by simple root-finding algorithms.
In commenting on this Answer the OP requests "the analytical solutions". This seems impractical. The function defined as inverse to $f(x)$ has no elementary form and is of no advantage since the numerical solution can be easily computed to any desired precision. The constants and parameters of this problem are not given to any particularly high precision, so carrying out the corresponding solution's precision to exceptional accuracy would seem pointless.
Likely we can solve equations VII and XIII for $x+y$ in a manner similar to how above we solved equations V and XII for $x$. If $y = (x+y) - x$ turns out to be positive, then also $T_L \gt T_H$ can be attained. After a unique value $T_I$ is obtained from equation VIII, we are done solving once we check to see if it satisfies the inequality:
$$ T_L \gt T_I \gt T_H $$
