How do we know that force is a vector? With displacement, our axoims and 'laws' of vectors work because that is how we designed vectors to work, as it seems obvious that displacement would follow the 'triangle law of vector addition', but how do we know that forces can also be modelled by vectors? Moreover, I do not understand how we are able to resolve a force vector into its 'horizontal' and 'vertical' components (using trigonometry) and treat them independently of each other; I understand that if force is a vector, we can add forces according to the law of vector addition, but what allows us to resolve a vector into two parts, and then pretend as if that is what is actually happening in the system? 
 A: Well, first of all, it is clear that time derivatives of vectors are vectors.  An easy way to see this physically is by thinking about the leading terms of a Taylor series:
$$
\vec{x}(t_0 + \Delta t) = \vec{x}(t_0) + \Delta t \left.\frac{d\vec{x}}{dt}\right|_{t=t_0} + \cdots
$$
So, we know $\Delta t$ is a scalar thus $d\vec{x}/dt$ must be a vector as that's the only thing you can add to a vector.
Well, let's assume that Newton's second law is true, and also that displacements are vectors.  So now we have
$$
\begin{align}
\vec{F} &= m\vec{a}\\
 &= m\frac{d^2\vec{x}}{dt^2}
\end{align}
$$
So, if displacement is a vector and Newton 2 is correct, force is a vector, since it is proportional to the second derivative of displacement with respect to time.  (This is true even if mass is not constant, since any time derivative of mass must be a scalar, which follows by a similar argument to that above.)
So either you don't believe one of these steps, or you accept that force is a vector.
I don't completely understand your second problem.  Once you have a vector then you can just crank up the whole mechanism of vector spaces on it, and in particular you know that you can resolve it into components with respect to a set of basis vectors, that that resolution is unique, that you can add things component-wise and so on.  So, for instance
$$
\begin{align}
\vec{u} + \vec{v} &= \sum\limits_i u_i\vec{e_i} + \sum\limits_i v_i \vec{e_i}\\
 &= \sum\limits_i (u_i + v_i)\vec{e_i}
\end{align}
$$
Where $\left\{\vec{e_i}\right\}$ are a set of basis vectors.  This is something that you can show is true for vector spaces, and so you can just use the mechanism & the maths will work.  Whether that is 'what is actually happening' I think is a question for philosophy not physics: for physics it is enough that the maths works.
A: A more philosophical answer to your question...
Physical laws (or in this case concepts like "force") can only be proven if you accept a certain set of axioms to be true. Whether what you are doing makes sense is eventually only confirmed by comparison with experiments. This process is also called "scientific method". 
So from a physicist's point of view the way of thinking is something like this:


*

*you want to describe some mechanical system

*you figure that it might be a good idea to introduce a certain vector quantity ("force") that satisfies some axioms (Newton's laws)

*you do calculations and compare the results with experiments

*you find that your model/theory agrees with experiments (within limits of applicability)


So basically, it just works if force is a vector. You are free to invent another theory where force is not a vector and see if you can make that theory work. Most likely it would either not work or the force in your theory would have nothing to do with what we would commonly associate with a force ("a push in one direction").

how do we know that forces can also be modelled by vectors? 

As above, you are not going to connect a mathematical thing like a vector to something physical (a force) without some additional input. If you look in tfb's answer, you could derive something if you accept Newton's second law to be true. Perhaps you could base it on some other physics axiom, but you are not going to proof it with mathematics only.
Your second part of the question is also known as Newton's 4th law or "principle of superposition" and is another axiom which you need to accept.
