Not really an full answer, but some comments (that hopefully answer some of your queries).
There seems to be a big confusion here : what do we want to integrate, i.e. to define $\int_{\mathbb{Z}_p} f(x)dx$, what is the 'nature' of $f$, and of the result ?
For the first one, (it's the one I am familiar with), the Haar measure on $\mathbb{Z}_p$ is in particular a map $\mu$ that assign to a Borel subset (say $E$) of $\mathbb{Z}_p$ real number $\mu(E)$. Take $f=1_E$ the characteristic function of $E$, then
$$\int_{\mathbb{Z}_p} 1_E d\mu=\mu(E) \in \mathbb{R},$$
by definition. The result is then a real number. That tells us that measure theory is about integration of functions with value in $\mathbb{R}$ !
That means that in this context, for example, $\int_{\mathbb{Z}_p} xd\mu(x)$ has no meaning from the point of view of this definition of integral.
If one is to look for an analogue of Riemann sums, we may notice that the sequence $(n)_{n\in \mathbb{N}}$ is equidistributed on $\mathbb{Z}_p$ with respect to $\mu$ (An ergodic theorist would say that the transformation $T:\mathbb{Z}_p\to \mathbb{Z}_p, T(x)=x+1$ is uniquely ergodic). The analogue of Weyl's criterion holds: for $f$ a real-valued continuous function, bounded on $\mathbb{Z}_p$, then
$$\int_{\mathbb{Z}_p} fd\mu=\lim_{N\to +\infty} \frac1N \sum_{n=0}^{N-1} f(n).$$
Another thing: the formula for integration on $\mathbb{Q}_p$ in the OP is in this case an analogue of the definition of the improper integral:
$$\int_{-\infty}^{\infty} fdx := \lim_{T\to +\infty} \int_{-T}^T f(x)dx,$$
which allows to make sense of functions whose integral in the measure theory sense has no meaning, like $f(x)=\sin(x)/x$.
About the Volkenborn integral, although this is not said, if we are to believe https://en.wikipedia.org/wiki/Volkenborn_integral, is made to integrate functions $f$ with values in $\mathbb{C}_p$ (but let's reduce it to functions with values in $\mathbb{Q}_p$ to be able to compare with the latter definition). The definition given in the OP
$$\int_{\mathbb{Z}_p} f(x)dx:=\lim_{N\to +\infty} \frac1{p^N} \sum_{n=0}^{p^N-1} f(n),$$
looks pretty much the same than the analogue of Riemann sums above, but with one big difference: one looks only at the subsequence $(p^N)_{N\geq 0}$. Indeed, an easy but instructive example is to compute these Riemann sums for $f(x)=x$. Then
$$\frac1{n} \sum_{k=0}^{n-1} f(k)=\frac{n-1}2 \in \mathbb{Q}_p,$$
which does not converge (recall the topology is the $p$-adic one), but does for the subsequence $(p^k)_{k\geq 0}$, to $-1/2$. This gives us $\int_{\mathbb{Z}_p} xdx=-1/2$, as said in the above wikipedia link (which contains, btw, a few formulas as required).
The third definition deals with $\mathbb{Q}_p$-linear vector space homomorphism of locally constant functions to $\mathbb{Q}_p$. So clearly here we are also looking at integration of function $f:\mathbb{Z}_p\to \mathbb{Q}_p$. The Volkenborn integral is an example, locally constant functions being strictly differentiable. But it's not the only one (EDIT : I previously stated that the Volkenborn integral is invariant by translations, which was wrong, as noted by Dap). So this definition is more general (the dirac measures works, for example).
I hope this clarifies a little bit...
Several years ago, I directed an informal undergraduate reading course on $p$-adic numbers. We followed Neal Koblitz's book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions (GTM volume 58). I really like the way this book presents the subject, and the students had a similar level of background to what you describe and found the book to be at an appropriate level, so it's worth considering this book if you want a written reference for the course.
Koblitz presents the $p$-adic numbers primarily through the analytic construction. The algebraic construction of the $p$-adic integers isn't explicitly introduced in those terms, but the essence of it is contained in the following theorem that Koblitz proves in chapter 1:
Theorem 2. Every equivalence class $a$ in $\mathbb{Q}_p$ for which $\lvert a \rvert_p \leq 1$ has exactly one representative Cauchy sequence of the form $\{a_i\}$ consisting of integers such that $0 \leq a_i < p^i$ and $a_i \equiv a_{i+1} \pmod{p^i}$ for all $i$.
The proof of this theorem is followed by a discussion of $p$-adic expansions. I don't think Koblitz explicitly describes this as a separate construction using the language of abstract algebra (that is, viewing these integers as elements of $\mathbb{Z}/p^i\mathbb{Z}$), but you could add it as a remark that immediately follows from that theorem and the discussion afterward.
In other words (whether or not you use this book), I would suggest using the analytic construction to prove the existence and basic properties of $p$-adic expansions of $p$-adic numbers first (and to favor the representation of $p$-adic expansions using sequences of integers rather than sequences of residue classes of integers), and only afterwards remark that in fact one could use this as an alternative algebraic construction of the $p$-adic numbers.
This is actually fairly consistent with Hensel's original motivation, which was to explore the analogy between power series in function fields (e.g. Taylor series of complex-analytic functions at a point) and power series in number fields (that is, "$p$-adic expansion of an algebraic number"), and to use this to prove theorems in algebraic number theory. The concept of a $p$-adic expansion is central. See here for some references on the history of the $p$-adic numbers.
Best Answer
Yes, $\mathbb{Z}_p$ is a closed subset (it is the closed ball of radius $1$).
To answer your second question, this does mean that $\sqrt{2} \in \mathbb{Z}_7$. Another way to see this is that $\mathbb{Z}_p$ is a PID, hence a UFD, and so integrally closed in its field of fractions $\mathbb{Q}_p$. The equation for $\sqrt{2}$ is $x^2 - 2 \in \mathbb{Z}_p[x]$.