There are two parts of your proof that I would consider incorrect, leaving out important details (which I'm sure you were aware of, but didn't write down):
In the $P \implies Q$ direction, Case 2, you only show that $\deg v_0 \ge 2$, which doesn't mean that it's even: $17 \ge 2$ and $17$ is odd. What you really want to argue here is that we use edges out of $v_0=v_k$ once at the very beginning, once at the very end, and twice every time it appears in the middle of the walk (this part is identical to Case 1), for a total of $2n+2$ edges, which is even.
In the $Q \implies P$ direction, when you write
we branch into $w_j$, walk until we reach $u_i$ again and continue the walk
this will not generally produce an Eulerian tour at the end. The closed walk $w_j$ might visit $u_i$ many times, and if you continue along $c$ the first time you return to $u_i$, you might miss out on many edges in $G_j'$. Instead, what you should do in this step is walk the entirety of the closed walk $w_j$, starting at $u_i$ and therefore ending at $u_i$, and then continue along $c$ to $u_{i+1}$.
Aside from that, I just have a couple of comments on clarity.
First, when you begin your induction
Let $P(n)$ be the proposition that if all $v\in V(G)$ have even degree, then a connected graph $G$ has an Euler circuit. We induct on $n=|E(G)|$.
you're not really using the right form for $P(n)$: you haven't made it a statement that depends on $n$. What you really want to define $P(n)$ to be is the proposition that "If $G$ is a connected graph with $n$ edges and all $v \in V(G)$ have even degree, then $G$ has an Euler circuit." (This also avoids the problem that, within the statement, you talk about $V(G)$ before you've said what $G$ is.)
Second, I'm not sure $P(3)$ is the right base case. I'd actually go with $P(0)$ (a graph with one vertex and no edges is connected, all vertices have even degree, and it has a trivial closed walk) because in your induction step, when you delete the cycle $c$, you may well end up with plenty of components that $P(0)$ should apply to. (And if we have $P(0)$, we can use it to prove $P(3)$: remove the cycle of length 3, and you have three components with $0$ edges.)
Third, a minor nitpick: when you're combining all the cycles together, you should only take the closed walk of $G_j'$ the first time you reach a vertex $u_i \in c$ that is a part of $G_j'$. (This is obvious if you know what you're doing, but it needs to be said if you want to describe a correct algorithm.)
But overall, well done! This is a complicated proof to write, with lots of fiddly details (which I didn't fully appreciate myself before I had to teach a recitation on it last semester), and you did a good job hitting all the important points. Don't let the length of my comments make you think otherwise.
It's always a spanning tree.
You probably already noticed this, but for completeness: the resulting graph is acyclic, because every cycle in the original graph has been destroyed. So we need to show that the result is still connected.
Another characterization of connectivity will be useful here: a graph $(V,E)$ is connected if and only if for every nonempty $S \subsetneq V$, there is a crossing edge: an edge between a vertex in $S$ and a vertex in its complement $V \setminus S$. So let's check this for the graph after deletions.
For a given set $S$, because our starting graph was connected, there are some crossing edges. Let $e$ be the lightest of these edges. I claim that the edge $e$ is never deleted, and so there is also a crossing edge in the graph we get at the end.
For $e$ to be deleted, we'd first have to find a cycle containing it. That cycle contains at least one vertex in $S$ and at least one vertex not in $S$. Following that cycle starting from $S$, at some point we leave $S$ - but then we have to come back to $S$ by a different edge. This can happen multiple times, but even if it only happens once, we see that the cycle contains at least two crossing edges: $e$, and some other edge $e'$ (and maybe others).
Since $e$ is the lightest crossing edge, it is in particular lighter than $e'$. So it is not the heaviest edge on this cycle, and will not be deleted when we consider this cycle. The same argument holds for every cycle containing $e$, so the edge $e$ will never be deleted.
In fact, the tree $T$ we get at the end is a minimum spanning tree.
To see this, take any other spanning tree $T'$. Let $e$ be an edge of $T$ not in $T'$. Adding $e$ to $T'$ creates a cycle, and deleting any edge of that cycle would create another spanning tree. Let's add $e$ and delete the heaviest edge of that cycle.
That heaviest edge is definitely not $e$, because $e$ is not the heaviest edge of any cycle. So we added $e$ to $T'$, then deleted an edge heavier than $e$. This means that we've reduced the total weight of $T'$: therefore, $T'$ is not a minimum spanning tree. Since some minimum spanning tree must exist, it can only be $T$.
Best Answer
Connectivity is equivalent to the property that there is a path between any two vertices. If we can show that the graph is bridgeless – no edge will disconnect the graph if removed – then we are done.
Suppose there is a bridge, i.e. an edge whose removal disconnects the graph. Since all the vertices of the original graph are of even degree, after removing this edge, we have two disjoint graphs, each with one vertex of odd degree (that was incident to the deleted edge) and the other vertices of even degree. However, this contradicts the handshaking lemma that the sum of degrees over all vertices must be even. Thus there is no bridge, and removing any edge will keep the graph connected, preserving the property that there are paths between any two vertices.