Question: What is the meaning of $E(\mathbb{Q}_p)$ where $E$ is an elliptic curve?

Question

Question: What is the meaning of $E(\mathbb{Q}_p)$ where $E$ is an elliptic curve?

Suppose $E$ is an elliptic curve over $\mathbb{Q}$. Then I understand what $E(\mathbb{Q})$ means, it is those points $(x,y)$ on $E$ where $x, y \in \mathbb{Q}$.

1. I have read that $\mathbb{Q}_p$ is the $p$-adic completion of $\mathbb{Q}$. But I don't know how to imagine / grasp the idea of $E(\mathbb{Q}_p)$.

2. My understanding is that it is the points $(x,y)$ on $E$ where $x,y \in \mathbb{Q}_p$. Is that correct? How would one find such a point anyway, for example find such a point when $E$ has the equation $y^2 = x^3 - x$?

3. Is there any relation between $E(\mathbb{Q}_p)$ and $E$ when viewed as an elliptic curve over $\mathbb{F}_{p^k}?$ I know that to study a curve $E$ sometimes we want to work over a finite field and using some techniques, one can find a point on $E$ modulo $p$ and then lift it up to $E$.

4. Lastly, it is a theorem that $E(\mathbb{Q}) / n E(\mathbb{Q})$ is finite. Is it also true when $\mathbb{Q}$ is replaced by $\mathbb{Q}_p$?

Thank you very much.