1. No, not necessarily.  It is hard to get a faithful geometric picture of a non-Archimedean space.  It *may* be helpful to have schematic approximate pictures in mind like in Daniel Litt's answer, but it is just as important to recognize the limitations of these pictures.  Speaking only for myself, contemplating the picture in Daniel's answer did not help me understand $p$-adic numbers: I was exposed to the picture offhandedly in a course I took as a college freshman, but it didn't make much sense to me until I studied the algebraic and metric properties of non-Archimedean fields more carefully (at a later time).  Pictures here are a form of intuition.  Having intuition is always helpful and at times indispensable, but importing others' intuition often does not work: you need to develop your own.

2. I would say no to this as well.  Of course you should understand what $\mathbb{C}_p$ means and how it is constructed, but in general thinking of algebraic structures element by element is not so useful.  By this I mean that rather than thinking of an element of $\mathbb{C}_p$ as a certain Cauchy sequence of elements in algebraic extensions of $\mathbb{Q}_p$ of varying degree, it is (almost?) as useful, and logically simpler, just to think of $\mathbb{C}_p$ as a complete, normed field containing a dense copy of the algebraic closure of $\mathbb{Q}_p$ with the (unique) extension of the $p$-adic metric.

3. Oh, yes.  You should definitely understand why the completion of the algebraic closure of the $p$-adic completion of $\mathbb{Q}$ is algebraically closed!  Of course, it's best if you can embed this fact into a general understanding of non-Archimedean fields rather than learning and memorizing an argument which shows exactly this.  For instance, in [these notes][1] I deduce (Corollary 22) the fact that the completion of a separably closed normed field is separably closed from **Krasner's Lemma**, which to me personally has become one of the most useful and meaningful parts of the entire theory.  Later on I show that a complete, separably closed field is necessarily algebraically closed (Proposition 27).  These are the right explanations *for me*, and I think they are good ones, but I'm not saying they need to be the right explanations for you.  Maybe something else speaks to you more than Krasner's Lemma.

4. Why are you lamenting your lack of understanding of $\mathbb{C}_p$ if you don't know how it is used?  (This is not meant to be rhetorical or combative: it's a sincere question.)  There are a lot of different answers in different areas of mathematics.  Moreover, for many people (and even some number theorists), the honest answer is that it is not used for anything in particular.  For instance, above I referred to some of my notes for a course I taught last spring on local fields and adeles.  From the perspective of those notes, the Henselian field $\overline{\mathbb{Q}_p}$ is just as good and perhaps more natural.  On the other hand, for some people going to $\mathbb{C}_p$ is not far enough: it is not **spherically complete**, meaning that the key property of a locally compact field like $\mathbb{C}$ or $\mathbb{Q}_p$ that a nested sequence of closed balls of radii decreasing to zero clamps down on (i.e., has common intersection) a unique point does not hold in general.  If you want to do serious $p$-adic functional analysis -- e.g. if you want things like the Hahn-Banach Theorem to hold -- then you want to work in $\Omega_p$, the spherical completion of $\mathbb{C}_p$.  But my guess is that the average working number theorist doesn't even know what $\Omega_p$ is, so it depends a lot on what you want to do.

<b>Added</b>: rather than leave the impression that I magically know more than the average working number theorist, let me say that I learned about spherically complete fields from Alain Robert's nice GTM on $p$-adic analysis.  I can't recall ever having needed to use them in my own work.  






[1]: http://math.uga.edu/~pete/8410Chapter3.pdf