Explanation behind a behavior observed for certain quadrics Having two quadrics, given as polynomials with real coefficients $f (x) = 0$, $g (x) = 0$; $x \in R^3$; $f (x) < 0$ marks the interior, $f (x) > 0$ marks the exterior, $f (x) = 0$ marks the boundary.
Then, in certain cases, I have observed that:
the two quadrics do not intersect $\iff$ $(f + g) (x)$ is a complex quadric.

For example, two spheres moving along $x$ axis:
$$x^2 + y^2 + z^2 - 2 \lambda x = 1 - \lambda^2$$
$$x^2 + y^2 + z^2 + 2 \lambda x = 1 - \lambda^2$$
$$x^2 + y^2 + z^2 = 1 - \lambda^2$$
The spheres are not in contact for $|\lambda| > 1$, according to our expectations.

A parabola orientated and moving along $z$ axis and a parabola orientated and moving along $y$ axis:
$$x^2 + y^2 - z + \lambda = 0$$
$$x^2 + z^2 - y + \lambda = 0$$
$$2 x^2 + (y - \frac {1} {2})^2 + (z - \frac {1} {2})^2 + 2 \lambda - \frac {1} {2} = 0$$
The parabolas are not in contact for $\lambda > \frac {1} {4}$, according to our expectations.

A cone orientated along $z$ axis and a sphere moving along $y$ axis:
$$x^2 + y^2 - z^2 = 0 $$
 $$x^2 + y^2 + z^2 - 2 \lambda y = 1 - \lambda ^2$$
$$2 x^2 + 2 (y - \frac {\lambda}{2}) ^2 = 1 - \frac{\lambda^2}{2}$$
The cone and sphere are not in contact for $|\lambda| > \sqrt {2}$, according to our expectations.

What I would like to know is the general underlying theorem for this behavior, what conditions do $f (x)$ and $g (x)$ have to meet, so I can use this safely.

If we have quadrics cut out by $f$ and $g$, then $f + g \in (f, g)$ so $f + g$ vanishes on the intersection of $V(f)$ and $V(g)$.  There's nothing special about $f + g$ in particular here; it just happens that in the examples you chose that happens to be the thing that reveals something about what's going on.  In other cases you might want to take $f p + g r$ for some other polynomials $p, r$.
The natural setting for everything here is complex projective geometry.  If you have two irreducible surfaces in $\mathbb{CP}^3$, like your pairs of quadrics here, then they are either the same or they intersect in a (possibly reducible) curve.  This curve is either defined over the reals or it isn't.  If it is, then when you just look at the real points you'll see a real curve of intersection.  If it's not, then at most you'll see a finite number of real points of intersection (e.g. your first example when $\lambda = 0$.)
Note that there's a slight subtlety because some of the intersections could happen at infinity.
Unfortunately it's not easy to state a simple version of this that you can just apply.  You'd need to get a feel for complex projective geometry and some basic intersection theory.
