Hello and welcome to the site. I will try to give an answer to some of your questions and hopefully the remaining ones will be dealt with, or if not, you can still ask in the comments.
First, some introductory remarks: The state $|\psi>$ (I prefer this notation rather than $|\psi(x)>$ at least in the braket) is a vector, living in a complex vector space. I like to think that the vector $|\psi>$ contains all sorts of information about the state of the quantum system under examination. The term abstract perhaps refers to the fact that the vector representation, despite the fact that contains all the information about the physical quantum system at hand, does not concretely allow you to access these pieces of information (at least in an immediate/actual/concrete way).
Let us take for example the position representation, denoted by $\psi(x)$. This representation is obtained by projecting the state onto an element of the posiiton basis $|x>$, a basis that belongs in the vector space mentioned earlier. This is hence a concrete way of obtaining information about the position of the quantum system at some certain point in time, since the projection $\psi(x)=<x|\psi>$ is indeed the probability amplitude for the system to be located at point $x$ in space. Note that the set of vectors that are in one-to-one correspondence to each possible position the quantum system might be located at, indeed comprises a basis:
$$1=\int dx |x><x|,\hspace{2em} \delta(x-x')=<x|x'>$$
Equivalently, one can consider the momentum basis of the space. A representative of such a basis could be a vector of the form $|p>$ and hence the wave function in momentum space is given by $\tilde{\psi}(p)=<p|\psi>$ (tilde to distinguish from $\psi$). In this way, we obtain information about the momentum of the system by projecting onto the momentum basis, since $\tilde{\psi}(p)$ is the probability amplitude of the system having momentum $p$.
Now, to answer (1), $|\psi>$ is a column since you can represent vectors as columns. Similarly, its dual $<\psi|$ can be thought as a row. (2) is a more difficult question to answer. However, since $|\psi>$ is a vector inside a vector space and we have just discussed about two of the bases in that vector space, one can think of an analogue (finite dimensional): imagine the a 2D coordinate system and the basis {(1,0),(0,1)} and $\{(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}),(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})\}$. What I have done is to write down two different bases, according to which I can write any vector in the 2D plane as a linear combination of either of them (this would be our case if there were only two allowed values for both the momentum and the position of the quantum system). For more on that I refer you to (Are the position eigenkets $\lvert x \rangle$ really a basis for the space of states?, Ambiguity in number of basis vectors, Hilbert space of harmonic oscillator: Countable vs uncountable?, Conceptual difficulty in understanding Continuous Vector Space). For (4), according to my way of viewing things, I like to think that it is called abstract representation because it does not provide the information it (secretely) holds in an immediate way, unless you project it onto some basis. And as far as (5) is concerned, I do not suppose that you are to understand things by assigning some sort of picture to a vector representing the quantum state of your system.
I hope my answers do not confuse you and I hope I haven't told you anything wrong.
