Could any forces from the moon, the planets or the sun in orbit hypothetically influence seismic events on earth? And if yes how to approximately calculate and compare the magnitude of the forces?
EDIT: On the How Stuff Works website they actually mention the possible influence of the moon and the sun on the earth's crust. But what about the possible influence of other planets in the solar system and their interactions, with earthquakes?
The planets do not affect Earth seismicity, but the Sun and Moon do.
Earthquakes are caused by strains in the Earth's crust, and the gravitational strains produced by the planets, Moon, and Sun on the Earth's crust are tidal effects which fall off as the cube of the distance. Only the Sun (because of its large mass) and the Moon (because it is so close) produce significant tidal affects on the Earth. The tidal effect of Venus, the planet that comes closest to the Earth is 4 orders of magnitude less than that of the Sun and Moon. The tidal effect of the most massive planet, Jupiter, is 5 orders of magnitude less. Any claim that planetary positions can be used to predict earthquakes is unfounded.
The tides caused by the Sun and the Moon can affect the likelihood of an earthquake, but the effect is small, e.g. less than 1% according to one study. For earthquakes in or near ocean basins, both solid earth tides and water tides contribute. (The weight of water tides can induce stresses in the ocean bottom that can be an order-of-magnitude larger than the direct solid-Earth tidal effect.)
Tidal stresses are 3–5 orders of magnitude less than the stress relieved in typical earthquakes, so the tide can only trigger an earthquake that was likely going to happen soon anyways. Studies have claimed that high tidal stress can significantly affect the timing of shallow thrust earthquakes or make it more likely that an earthquake will grow to a larger size.
The idea is that tidal deformation (gravitational gradients) can increase strain, thereby triggering strain-relief: a quake.
In the extreme case of the Earth passing inside the Roche limit of a larger ($M$) celestial body, the answer is clearly "yes".
For instance, a rogue black hole would produce a gradient large enough to induce The Rapture (people on the line of syzygy would be lifted off the surface on both sides of the planet, those transverse to that would be pulled down into the crumbling crust, for a red hot molten demise...have I read the somewhere before?).
In more mundane circumstances, tidal gradients go as:
$$ \frac{ M }{R^3} $$
The sun competes with $M$, while the moon wins "out" on $R$. From there, look at the planets and see their relative contribution.
Being a tensor alignment, direction doesn't matter, so if Venus and Jupiter line up: they add, even though they are on opposite sides of the planet.
You just have to plug in the numbers.
Edit (based on comments):
With a gravitational potential $U(\vec x)$, the Hessian is:
$$ J_{ij} = \frac{\partial^2 U}{\partial x_i \partial x_j} $$
The tidal tensor is the symmetric trace free (aka "natural rank-2 tensor"):
$$ \Phi_{ij} = J_{ij} - \frac 1 3 {\rm Tr}(J) $$
which has 5 components. These induce quadruple deformations in the Earth's shape that are characterized by the 5 (real) spherical harmonics with $l=2$... e.g. prolate stretching. Because it's traceless, the longitudinal stretch is twice the 2 horizontal squishes, so volume is preserved. (Remember, the field lines converge so they don't point in the same direction on opposite sides of low tide).
One can liken this to the (dipole) polarization of an atom in a (vector) field, where the deformation is characterized by the $Y_{l=1}^{m \in (-1,0,1)}$ spherical harmonics.
It's all very geometric.
