I think I can safely say that nobody understands quantum mechanics.
Richard Feynman, in The Character of Physical Law (1965)
That said, I think I can attempt to clarify some of your misunderstandings
from my own understanding. In fact someone set me straight if I have any
issues of my own :)
The entire notion of a point particle is essentially a classical
approximation (as far as geometry goes). In fact, all the spatial information
that can be known (ie not completely transparent to the rest of the universe)
about a particle is completely contained in its wave function, complex valued
defined at every point in space. But the wave function in time must satisfy
equation, and it has been shown by people smarter than I that wave
function solutions *must* satisfy the uncertainty principle.
Its easiest to consider what this means in one dimension. Solutions of
the Schrödinger Equation are linear combinations of sinusoidal functions of
all wavelengths and velocities (with the solution for a particular particle
determined like with any differential equation by the spatial and temporal
boundary conditions). This is immediately consistent with the wave description
of light and matter, as a sinusoidal function has a definite velocity but
its position is not defined at all (it looks like a wave :P). So how then
can we get a localized particle, like those we apparently observe enough
to create an entire classical theory around? Well it turns out that taking
linear combinations of waves of differing velocities causes local areas of
destructive and constructive interference, and one can mathematically construct
what's known as a wave
packet. Btw, the time evolution of the wave packet in the picture
on wikipedia is incorrect for solutions to the Schrödinger Equation: particle
wave packets necessarily disperse over time depending on the represented wave
velocities (don't quote me on that). This means the range of represented wave
velocities actually has physical significance. Anyway there's a limit to how
localized a wave packet can get, called a Gaussian wave packet. To achieve
this limit, one has to sum over essentially every possible wave velocity.
So solutions of the Schrödinger Equation can be something with no
localization at all and a perfectly well defined velocity, like a sinusoidal
function, or with a very acute (but not perfect) localization achieved by
an almost infinite range of velocities of component waves. In fact there is a very simple
expressing the relationship between the smallness of the localization to
the range of velocities (momenta, actually)...
So all that's not that bad. The real strangeness of QM comes with what
observation does to the wave function of a particle. Somehow, the act of observation (something I am not knowledgeable enough to define, but examples of which are hitting it with a photon or having it excite the screen in the double slit experiment, or even covering up a slit thus knowing it must go through the other) "collapses" the wave function of a particle back into its most localized form. The probability distribution of the center of the new localized form is given by the product of the wave function with its complex conjugate just before the observation.
The interference pattern corresponds to the probability distribution of particles when they reach the screen behind the double slit. If I fired only one particle through the double slit, it would cause a single photon (probably) to be emitted from the screen, with its location determined by the probability distribution. We can see an interference pattern because we are firing a beam of particles, not just one at a time. The kicker from the experiment is if we observe the interference pattern (say by collecting billions of data points from electrons fired one at a time), information about which slit each electron went through (in particular) is transparent to the rest of the universe; it cant be determined from say where that electron struck the screen. This seems to be where your misunderstanding lies, at least in part. If we had observed which slit the electron went though (by covering up a slit, or by shining light on it and doing what scientists do), then the interference pattern would have disappeared.