No, it is not possible with that information. Here is an easy counterexample:
Suppose that $d(A,C)=d(B,C)=r$. Then if we take the point $C$ as a center of some circle with radius $r$ then the points $A$ and $B$ can be anywhere on the circle so obviously their distance is not determined. You need additional information, for example, an angle between sides $AC$ and $BC$ and then you have a triangle which have the length of the third side $AB=d(A,B)$ determined by the length of the sides $AB$ and $BC$ and the angle between them.
With that angle you need only one measurement but without an angle the counterexample shows that it is not possible, even with many measurements, because $d(A,B)$ will be impossible to determine because the first measurement does not determine it and because there is no measurement that determine it, no matter the number of measurements.
But you can know something of course. suppose $d(A,C)=r_1$ and $d(B,C)=r_2$ and suppose, without losing generality that $r_2>r_1$. Then if we take $C$ as the center of the circles with radii $r_2$ and $r_1$ then you have $r_2-r_1\leq d(A,B)\leq r_2+r_1$ (draw picture).
What is the optimal check for collinearity to make sure that it is actually possible to define a quadrilateral using these four points?
I don't know about optimal, but the 2D analog of cross product (the test OP is using) is probably the simplest and most robust check to implement for numerical computation. In detail, if $\epsilon$ is the largest positive value you consider zero (machine epsilon), then the three points $(x_a, y_a)$, $(x_b, y_b)$, and $(x_c, y_c)$ are collinear if and only if
$$\lvert (x_b - x_a)(y_c - y_a) - (x_c - x_a)(y_b - y_a) \rvert \le \epsilon$$
Computationally, this "costs" two multiplications, five subtractions, and either two comparisons or one comparison and one absolute value function; not much at all.
If the Euclidean distances between the points are already known, then triangle inequality could be used. However, the square root operation is usually much more costly ("slower") computationally than multiplication, and the inequality does not hold for squared lengths, so it is not very attractive for this particular case, where we only know the point coordinates.
[Are my checks] enough or do I have to add something else?
There are four ways to choose three out of four unique items (see binomial coefficient at Wikipedia):
$$4 \text{ choose } 3 = \binom{4}{3} = 4$$
This means that if you have four points, you need four tests to verify that no three of them are collinear.
In other words, checking collinear(p1,p2,p3), collinear(p2,p3,p4), and collinear(p1,p3,p4) is not sufficient -- it misses the case where points 1,2, and 4 are collinear -- and you need to also check for collinear(p1,p2,p4).
The test mentioned earlier will also catch the case when two or more points are at the same coordinates, so there is no need to test for equality.
Here is an example of the test in pseudocode. It returns True if at least three of the four points are collinear:
Function Collinear3of4(p1, p2, p3, p4, eps=0.0000005):
# (p1, p2, p3) are collinear if and only if
# abs( (p2.x-p1.x)*(p3.y-p1.y) -
# (p3.x-p1.x)*(p2.y-p1.y) ) <= eps
(x12, y12) = (p2.x - p1.x, p2.y - p1.y)
(x13, y13) = (p3.x - p1.x, p3.y - p1.y)
(x14, y14) = (p4.x - p1.x, p4.y - p1.y)
(x23, y23) = (p3.x - p2.x, p3.y - p2.y)
(x24, y24) = (p4.x - p2.x, p4.y - p2.y)
# Test each unique triplet.
# 4 choose 3 = 4 triplets: 123, 124, 134, 234
Return ( (Abs(x12*y13-x13*y12) < eps) Or
(Abs(x12*y14-x14*y12) < eps) Or
(Abs(x13*y14-x14*y13) < eps) Or
(Abs(x23*y24-x24*y23) < eps) )
In C, the temporary variables could/should be considered premature optimization, as the compiler should be able to do the corresponding optimizations automatically. In Python, member access is slower than local variable access and the documentation recommends using temporary variables over repeated member accesses, so the temporary variables are justified -- not to reduce the number of subtractions, but to use local variables instead of member accesses. In C++, the temporary variables are justified if the members are functions (like in Qt point types); otherwise, the optimization is premature. (C++ compilers should be able to optimize even the member function calls, if they are pure getters, actually.)
In fact, I personally would only accept a version of the function if it has comments making the math it implements clear, and, if it uses any, the temporary variables are well named.
If the temporary variables were named e.g. xA, yA, and so on, it would be near impossible to tell if the math part is correct or not. With these names, one can easily verify if all four sets (123, 124, 134, 234) are covered.
This all loops back to optimal, you see. I value robustness and maintainability: I don't want to, or ever try to remember any details of the code I wrote a year, a month, or even a week ago. I want the code to tell me what it does, and the comments to tell me why it does it, and what the purpose of the code is.
(No, my comments are not nearly as good as I'd like them to be, and it is one of the things I need to work on myself.)
Best Answer
A closed curve is a circle with points $a,b$ as diameter if and only if all points $c$ on the curve satisfy $d(a,c)^2+d(c,b)^2=d(a,b)^2$.