Does DTW return smaller distance measure than Euclidean Distance? QUESTION 1: When computing the distance between two time series, shouldn't the DTW distance measure return a smaller distance than the Euclidean distance (assuming DTW internally uses the Euclidean Distance (ED))?
In my understanding, the DTW algorithm will choose the index $i$ of a time series $T$ that best matches a given index $j$ of the other series $Q$. 
This should only yield a better result than merely taking $i = j$ like the Euclidean distance, shouldn't it ?
I know that there are a few constraints on the DTW algorithm, but in my mind, these do not explain why the distance is greater in DTW than in ED.
I've been trying to find a bug in the code but the library I'm using seems solid. I'm trying to do similarity search and using both separately yields the same results on artificial datasets however, I would like to compare the two algorithms on real data. If DTW yields better results, I could use an approximation like FastDTW to overcome the $\mathcal{O}(N^2)$ time and space complexity of DTW and have the same complexity as ED $(\mathcal{O}(N))$.
QUESTION 2: If comparing the distances is not the right option, how do I compare the two algorithms (apart from visually) ?
Have a great day, thanks in advance !
 A: I know it's a little late, but I can answer your first question.
First, let's denote the distance between two points $a$ and $b$ as $d(a,b)$. In your case, $d$ will be ED.
Next, let's denote the "DTW distance" between $T[i]$ and $Q[j]$ as $\delta(T[i], Q[j])$. Note that I am using different symbols to denote the two distances.
In DTW, for $i>0$ and $j>0$,
$$ 
\delta(T[i], Q[j]) = d(T[i],Q[j]) + min(\delta(T[i-1],Q[j]), \delta(T[i],Q[j-1]), \delta(T[i-1],Q[j-1]))
$$
For $i=0$ or $j=0$, $\delta(T[i], Q[j])$ is defined a little differently, but with the same intuition.( see https://en.wikipedia.org/wiki/Dynamic_time_warping#Implementation )
Finally, the distance that the algorithm returns is $\delta(T[I],Q[J])$, where $I$ and $J$ are the endpoints of the two time series' $T$ and $Q$.
As you can see, the DTW algorithm actually computes the distance in a linear manner, very much like Manhattan Distance. The metric you defined ($d$=Euclidean Distance in your case) is only used to compute the pointwise distances.
In short, comparing ED with DTW(with distance metric as ED) would be like comparing $ED(T,Q)$ to $\Sigma_{a,b}ED(a,b)$, where $a$ and $b$ are optimally matched, so it's not a fair comparison.
For Question 2, I don't understand what you are asking.
