The factor $e^{-1/t}$ is $\asymp1$ for $t>1$, and so, it may be dropped. So, the integral in question is finite iff $I_j<\infty$ for all $j=1,\dots,6$, where
\begin{equation}
I_j:=\iiint\limits_{R_j}\frac{dx\,dy\,dt}{x^a\,y^b\,(x+y)^{c_1}\,(x+y+t)^{c_2}\,t^{1/2}},
\end{equation}
\begin{align}
R_1&:=\{(x,y,t)\colon 1<x<y<t\}, \\
R_2&:=\{(x,y,t)\colon 1<y<x<t\}, \\
R_3&:=\{(x,y,t)\colon 1<x<t<y\}, \\
R_4&:=\{(x,y,t)\colon 1<y<t<x\}, \\
R_5&:=\{(x,y,t)\colon 1<t<x<y\}, \\
R_6&:=\{(x,y,t)\colon 1<t<y<x\}.
\end{align}

We have
\begin{equation}
I_1\asymp\int_1^\infty\frac{dx}{x^a}\int_x^\infty\frac{dy}{y^{b+c_1}}
\int_y^\infty\frac{dt}{t^{c_2+1/2}},
\end{equation}
so that $I_1<\infty$ iff
\begin{equation}
c_2+1/2>1,\quad b+c_1+c_2>3/2,\quad a+b+c_1+c_2>5/2.
\end{equation}
Similarly, $I_2<\infty$ iff
\begin{equation}
c_2+1/2>1,\quad a+c_1+c_2>3/2,\quad a+b+c_1+c_2>5/2.
\end{equation}

The integrals $I_3,\dots,I_6$ are treated similarly. As the result, the integral in question is finite iff
\begin{equation}
c_2+1/2>1,\quad a+c_1+c_2>3/2,\quad b+c_1+c_2>3/2,\quad a+b+c_1+c_2>5/2.
\end{equation}