Why matrices are multiplied the way they are multiplied? I couldn't see a specific reason for multiplying every row of A with every column of B. Is this an arbitrary property of multiplication function of matrices?
Instead, why don't we simply multiply row#1 of matrix A with row#1 of matrix B, which would help us to do multiplication easier without all that confusion of what we are gonna multiply with what? In this case, product of A(mxn) and B(kxn) would be P(mxk). 
I know a lot of things would change in today's mathematics if we define multiplication this way. But would that approach cause any problem in the future?
 A: If we consider matrices simply as tables of numbers than we can define many possible different binary operations that we can call '' multiplications'', simply using this name to distinguish this operation from the addition (defined as the sum of corresponding elements). Obviously different definitions give different properties of the ''multiplication'' and someone can be useful in some contest, but not in other.
As an example the Hadamard product of two matrix (defined as the product of the corresponding elements) is associative, distributive and also commutative, but can be defined only for matrices that have the same dimension, and ( as far as I know) is used in computer graphic.
The Kroneker product is another possible kind of multiplication, that has usefull properties and has important applications being related to the tensor product of linear transformations.
The usual row-column product  has the advantage that it can represent the action of linear transformations between vector spaces, and capture all properties of these transformations (linearity, associativity, non commutativity, existence of a neutral element and of not invertible elements). There is some amount of convection in the definition,  in the sense that we can chose the row to the left and column to the right ( as usual) or vice versa, but really these two possible alternative give isomorphic structures.
A: Suppose you have $x,y,z$ defined in terms of $p,q$, let's say, $$\eqalign{x&=3p+4q\cr y&=5p-2q\cr z&=-p+7q\cr}$$ and you have $p,q$ defined in terms of $a,b,c,d$, let's say, $$\eqalign{p&=4a-3b-c+2d\cr q&=9a+5b-6c+3d\cr}$$ and you want to express $x,y,z$ in terms of $a,b,c,d$. Well, all you need to do is extract the matrices of coefficients, and multiply them: $$\pmatrix{3&4\cr5&-2\cr-1&7\cr}\pmatrix{4&-3&-1&2\cr9&5&-6&3\cr}$$ The product will give you the coefficients in the expressions for $x,y,z$ in terms of $a,b,c,d$. 
See also Why, historically, do we multiply matrices as we do?
