When you divide through by 3, the resulting congruence should be
$$15x\equiv 5\pmod{26},$$
not
$$15x\equiv 3\pmod{26}.$$
The diophantine equation $ax+by = r$ can be solved if and only if $\gcd(a,b)\mid r$. In that case, the Euclidean algorithm provides an effective way of finding all solutions.
The key to this is that the collection of all possible values of $ax+by$ is precisely the set of all multiples of $\gcd(a,b)$.
Now consider $ax+by+cz = t$. This is equivalent to solving $\gcd(a,b)w + cz = t$, since any solution to the latter yields a solution of the former (by suitable choice of $x$ and $y$ so that $ax+by = \gcd(a,b)w$) and conversely, any solution to $ax+by+cz=t$ yields a solution to $\gcd(a,b)w+cz = t$. Thus, the diophantine equation $ax+by+cz=t$ is equivalent to the diophantine equation $\gcd(a,b)w+cz=t$, which can be solved if and only if $\gcd(\gcd(a,b),c) = \gcd(a,b,c)$ divides $t$.
Similar considerations yield that the diophantine equation
$$a_1x_1+a_2x_2+\cdots+a_kx_k = d$$
with $k\gt 0$ has a solution if and only if $\gcd(a_1,a_2,\ldots,a_k)$ divides $d$. When it does, the Euclidean algorithm provides an effective way of finding the solutions.
Gerry Myerson has already addressed the other part of your question.
Best Answer
Start by solving the equation $8s+13t=1$. You can do this by inspection, for $s=-8$, $t=5$ obviously works. Or else you can use the machinery of the Euclidean Algorithm.
So $x_0=(-8)(1571)$, $y_0=(5)(1571)$ is a solution of the equation we were given.
All solutions of that equation are given by $x=x_0+13t$, $y=y_0-8t$, where $t$ ranges over the integers.
Remark: We need not have started from a solution of $8s+13t=1$. The only reason we did it that way was to connect the solution with material that you have already covered.
If we find any particular solution $(x_1,y_1)$ of our equation, then all solutions are given by $x=x_1+13t$, $y=y_1-8t$.
If you have done Linear Algebra, or Differential Equations, you have seen this kind of thing before. We got the general solution of our equation by taking a particular solution, and adding the general solution of the homogeneous equation $8x+13y=0$.