Or best of all, can we actually walk through walls (tunneling). Is it really true that time slows down and I gain mass when I travel faster?
And actually we don't define a particle as a physical entity but a probability distribution function. A mathematical equation of probably of where a particle might be.
And some of the theories describing matter these days are pretty bizarre.
It is not "made" of either of those things. You are confusing two highly useful conceptual models of a thing with the thing itself. Matter is neither particle nor waves, and if you look at the axioms of quantum mechanics [For just one example (which is typical): http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Postulates_of_quantum_mechanics] You will not find any reference to particles, waves, wavefunctions, or duality. These are high school concepts, which are useful ... and wrong.
Can we actually determine the outcome of an experiment by simply observing it?
In one word: yes.
Quantum mechanics does not make the results of experiments indeterminate. At. All. It makes some outcomes probabilistic under some experimental conditions, which is not the same thing.
Is it really true that we cannot know the speed of a particle and its position.
Speed is a classical notion, so the answer to your question is: Yes. We can know the speed of a particle and its position.
In quantum mechanics, we have an operator for momentum, but not speed. There is NO observable corresponding to "speed." We have an operator for position. Conjugate momentum and position operators don't commute, so there is a lower bound on their simultaneous precision, which is ℏ/2. This is a well know result, mathematically provable for anyone who's taken his first course in quantum mechanics. There's nothing mysterious or weird about it.
Or best of all, can we actually walk through walls (tunneling).
No. You can't.
The probability of an object with kinetic energy T tunneling through a potential energy barrier of energy V with width L is e-2L√(2m(V-T))/ℏ For typical walking speeds and even very weak walls only a few centimeters thick, this probability is ≈ e-1033. So start walking into walls now, and you will be doing that for something on the order of 10 quadrillion times the lifetime of the universe without winding up on the other side even once.
Is it really true that time slows down and I gain mass when I travel faster?
Not for you it isn't. Neither your proper time nor your mass appear to change. Observers in some reference frames see your clock running slow and an increase in your relativistic energy. Most physicists really don't talk about "mass increasing" any more, because that idea is not really very conceptually useful. They would agree that if you're moving relative to them, your energy appears to be greater.
And actually we don't define a particle as a physical entity but a probability distribution function. A mathematical equation of probably of where a particle might be.
Again, you are confusing a model with the thing itself. A particle is not a probability distribution. There is a probability of measuring certain attributes of a particle based on its state vector. But the particle itself is NOT the state vector, that is simply a thing which describes the particle, and describes how certain operators -- called observables -- will act when we do measurements.
And some of the theories describing matter these days are pretty bizarre.
The theories are only "bizarre" if you're of the very arrogant opinion that the same rules applied to things 15 or 16 orders of magnitude smaller than your ordinary experience or 20 or so orders of magnitude larger than your ordinary experience will produce the kinds of outcomes you're familiar with. But the same laws govern your ordinary experience as govern those very large, or very small, or very quick, or very slow events. And there is nothing strange -- even in your everyday experience -- about those laws.