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Since there are already several very good answers, I just discuss question 4 (how is ${\mathbb C}_p$ typically used?) with one example of use which made a great impression on me when I learnt it, and made me think that ${\mathbb C}_p$ was something deep and serious, and not only a very amusing curiosity. This example is a theorem of Tate and Sen, which states that if $V$ is a finite-dimensional over $\mathbb{ Q_p}$ vector space with a continuous linear action of $G=$Gal$(\overline{\mathbb Q_p}/{\mathbb Q_p})$ (that is, $V$ is a $p$-adic representation of $G$), then the following are equivalent:

(1) $\dim_{\mathbb Q_p} (V \otimes {\mathbb C_p})^{G} = \dim_{\mathbb Q_p} V.$ (Here, G acts on $\mathbb{C_p}$ by extending by continuity its action on $\overline{\mathbb Q_p}$ and it acts on $V \otimes {\mathbb C_p}$ by acting on both factors.)

(2) The inertia subgroup of $G$ acts on $V$ through a finite quotient (in more knowledeable words, $V$ is potentially unramified).

To appreciate this theorem, it may be useful to solve for oneself the following elementary exercise: if in (1), ${\mathbb C}_p$ is replaced by $\overline {\mathbb Q_p}$, then (2) should be replaced by "$G$ acts on $V$ through a finite quotient". Somehow, replacing $\overline {\mathbb Q_p}$ by its completion allows (1) to see inside the group $G$ and detect the behaviour of the inertia subgroup in it.

I believe that someone who understands the proof of this theorem has necessarily a good understanding of $\mathbb{C}_p$, and this will be my answer to question 3 as well: knowing the proof of the basic assertions on $\mathbb{C}_p$ given in the questions is a first step into a good understanding of that field and its elements, but won't take you very far. Learning the proof of the above theorem will let you get a much deeper look inside $\mathbb{C}_p$ -- and in addition you will learn a nice result, which is a first step in the fundamental $p$-adic Hodge Theory.

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Since there are already several very good answers, I just discuss question 4 (how is ${\mathbb C}_p$ typically used?) with one example of use which made a great impression on me when I learnt it, and made me think that ${\mathbb C}_p$ was something deep and serious, and not only a very amusing curiosity. This example is a theorem of Tate and Sen, which states that if $V$ is a finite-dimensional over $\mathbb{ Q_p}$ vector space with a continuous linear action of $G=$Gal$(\overline{\mathbb Q_p}/{\mathbb Q_p})$ (that is, $V$ is a $p$-adic representation of $G$), then the following are equivalent:

(1) $\dim_{\mathbb Q_p} (V \otimes {\mathbb C_p})^{G} = \dim_{\mathbb Q_p} V.$

(2) The inertia subgroup of $G$ acts on $V$ through a finite quotient (in more knowledeable words, $V$ is potentially unramified).

To appreciate this theorem, it may be useful to solve for oneself the following elementary exercise: if in (1), ${\mathbb C}_p$ is replaced by $\overline {\mathbb Q_p}$, then (2) should be replaced by "$G$ acts on $V$ through a finite quotient". Somehow, replacing $\overline {\mathbb Q_p}$ by its completion allows (1) to see inside the group $G$ and detect the behaviour of the inertia subgroup in it.

I believe that someone who understands the proof of this theorem has necessarily a good understanding of $\mathbb{C}_p$, and this will be my answer to question 3 as well: knowing the proof of the basic assertions on $\mathbb{C}_p$ given in the questions is a first step into a good understanding of that field and its elements, but won't take you very far. Learning the proof of the above theorem will let you get a much deeper look inside $\mathbb{C}_p$ -- and in addition you will learn a nice result, which is a first step in the fundamental $p$-adic Hodge Theory.