I think there is no general rule for deciding when the error tends to zero. There are an infinite variety of cases - as you illustrate. You must examine the appropriateness of the approximation being made in each case.
Take your first example, which I will simplify to the 2D problems of finding the area and perimeter of a circle.
Approximate the area of the circle with horizontal strips which do not extend beyond the perimeter. As the strips are made narrower the total area of the strips increases hence the error decreases. In the limit of infinitesimally thin strips we get the exact value of the area.
Approximate the perimeter of the circle using horizontal and vertical straight lines, as done in the paradox in the Math SE question Is $\pi = 4$?
Start by approximating the circle with a square of perimeter 4. Then make indents by removing corners - the perimeter is still 4. As more and more corners are removed, the resulting staircase curve approximates more closely to the circle, yet the perimeter remains the same. The error never gets any smaller.
The correct way of doing this approximation is to use the hypotenuse of the triangles as the element of perimeter. While the length of the staircase curve never gets any smaller, the total length of the hypotenuses decreases towards the circumference of the circle.
In the Math SE answers it is pointed out that the staircase curve does not approach the circle smoothly. Smoothness is related to derivatives, so another way of putting this is that while two curves can be arbitrarily close at some point, their derivatives at that point can be significantly different.
The key to ensuring that the approximation gives the correct result when you integrate, is to examine the smoothness of the approximation. You can see that the staircase curve never gets any smoother - it is always jagged if you look at it on a small enough scale. So it can never provide an accurate approximation to the circle.
In practical terms examining smoothness of an approximation is doing the same as what you are asking a shortcut for - ie checking that the error tends to zero. I think there is no shortcut, no easy alternative to checking in each case that the approximation does in fact get closer to what it is approximating as the step size decreases. In many cases you will know from experience that it works. Otherwise you need to convince yourself that the error does decrease towards zero as the step size is reduced.
Your other examples :
While $ds=r\delta \theta$ works when the centre is the origin O (and $s=r\theta$ is in fact exact even for finite angles), it does not work when a point on the circumference is the origin. When OA is a diameter the relation $ds=r\delta\theta$ is good as $\delta\theta \to 0$, but as A moves closer to O then OA shrinks to zero and the approximation gets increasingly bad - as your hand-drawn diagram beings to show. Your guess ("Probably mistake : $r\delta\theta \ne ds$") is correct.
(i) Spring and block. You have expressed the spring force in terms of $y$, so your mistake is not failing to do so. Your mistake is failing to relate $dx, dy$ correctly : $dy=\sin\theta dx$. Then work done is $kxdx=kxdy/\sin\theta$. A diagram linking $dx$ and $dy$ will help you avoid this error.
(ii) PE of Chain. The height of the CM is $h=R(1-\cos\theta)$ so $dh=R\sin\theta d\theta$ not $R d\theta$. Your mistake is again that you have not drawn a diagram to relate $dh$ to $Rd\theta$.
(iii) Pressure in bowl. Not enough explanation of what you are doing. You seem to have ignored $l=2\pi R\cos\theta$.
$ds$ is the arc length. We know $ds/dt$ = speed = $|dr/dt|$ since the error tends to zero.
