Another condition "for a line to intersect another line, in 3D" is that there is a point that is on both lines. That sounds simplistic but that really is the condition.
Here is one way to solve your problem. The line that you want is in a plane that is parallel to the plane $3x-y+2z-15=0$. Any such plane has the equation
$$3x-y+2z+D=0$$
for some constant $D$. That plane must also go through the point $M(1,0,7)$, so we can substitute to find the value of $D$:
$$3\cdot 1-0+2\cdot 7+D=0$$
So $D=-17$ and the plane is $3x-y+3z-17=0$.
Combining that with your equations $\frac{x-1}{4}=\frac{y-3}{2}=\frac{z}{1}$ you can find the single point that is the intersection of that plane and your given line. The line you want goes through those two points.
You should be able to finish from here.
There are multiple ways to find the intersection of the plane and of the given line. You showed one way, by giving a parameterization of the line and solving for the parameter. You did make a computation error: the equation you get is
$$3(4t+1)-(2t+3)+2(t)-17=0$$
and the solution is
$$t=\frac{17}{12}$$
which makes $x=\frac{20}3,\ y=\frac{35}6,\ z=\frac{17}{12}$. Substituting this solution into all the equations checks, so this is the right answer. Thus the intersection point is
$$Q\left(\frac{20}3,\ \frac{35}6,\ \frac{17}{12}\right)$$
I got the intersection point by different means, by setting up and solving these simultaneous linear equations:
$$\begin{align}
\color{white}{1}x \color{white}{+0y}-4z&=1 \\
y-2z&=3 \\
3x-y+2z&=17
\end{align}$$
I got the same intersection point.
Best Answer
Equation of the line L1 in symmetric form is $$\frac{x}{1} = \frac{y - a}{0} = \frac{z - a}{0} \tag1$$ Equation of the Line L2 in symmetric form is $$\frac{x -a}{-3} = \frac{y - a}{-1} = \frac{z}{1} \tag2$$ Let the required line 'L' intersect L1 at P and L2 at Q.
Coordinates of P are $$(r,a,a) \tag3$$ Coordinates of Q are $$(-3s+a, -s+a, s) \tag4$$ Equation of L is $$\frac{x -r}{-3s-r+a} = \frac{y - a}{-s} = \frac{z-a}{s-a} \tag5$$ Given that this line L is parallel to the plane $x+y=0 \tag6$ Hence, its normal will be perpendicular to L So $$(-3s-r+a)+(-s)=0 \tag7$$ or $$r=a-4s \tag8$$ From equation 5 we have $$\frac{y-a}{z-a} = \frac{-s}{s-a}$$ Solving this for s, we will have $$s=\frac{a(y-a)}{y+z-2a} \tag9$$ From equation 5, we also have $$\frac{x -r}{-3s-r+a} = \frac{y - a}{-s} \tag10$$ Substituting the values of r and s obtained from equations 8 and 9, we get the required surface as $$y^2 + xy + yz + zx - 2ax -2az = 0$$