Is there any link between the sign of RHS in (3) and the bisector of the smallest (biggest) angle?
No, there is not.
Suppose there is - for example let the "+" sign determine the bisector of the smallest angle, so its equation is
$$\frac{a_{1}x+b_{1}y-c_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}}}= \color{red}+ \frac{a_{2}x+b_{2}y-c_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}}}\tag{3'}$$
Now, let's rewrite your (original) equation for the second line
$$a_2x+b_2y=c_2 \tag{2}$$
as
$$(-a_2)x+(-b_2y)=-c_2 \tag{2'}$$
This is an equation of the same line. Substituting its coefficients ($-a_2, -b_2, -c_2$) into $(3')$ we obtain
$$\frac{a_{1}x+b_{1}y-c_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}}}= \color{red} - \frac{a_{2}x+b_{2}y-c_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}}}\tag{3''}$$
Oops! $(3')$ and $(3'')$ for the bisector of the smallest angle - and for the same lines!
(The point is that equations $(1)$ and $(2)$ are not canonical - there are many such equations for the same line.)