Axioms for constructive Euclidean geometry
Have a look at Hartshorne's Geometry: Euclid and Beyond. He uses Hilbert's axioms for geometry and discusses (section 11) the following "circle-circle intersection axiom (E)":
Given two circles $\Gamma,\Delta$, if $\Delta$ contains at least one point inside $\Gamma$, and $\Delta$ contains at least one point outside $\Gamma$, then $\Gamma$ and $\Delta$ will meet.
Note that the hypothesis of this axiom is (at least arguably) a different way to say "unless they obviously don't" which is nevertheless based on betweenness and order rather than distance.
Hartshorne shows (section 12) that essentially all of Euclidean geometry can be developed in a plane satisfying the two-dimensional version of all of Hilbert's axioms, but with the completeness axiom replaced by the above axiom (E). This answers your second question: completeness is not needed, all you need is (E). Finally, in section 16 Hartshorne shows that your intuition about the algebraic version of (E) is correct: the cartesian plane over an ordered field satisfies (E) iff all positive numbers have square roots.
I think that the Alexandrov's system is an answer.
Check "Minimal foundations of geometry" by Aleksandrov (Russian original and there is a translation in Siberian Math. J. 35 (1994), no. 6, 1057–1069).
He also wrote a textbook for school students based on this system.
The set of geometric tools that are used in Euclidean geometry consists only of a pencil, ruler, and compass. Using this toolset, we cannot construct all points on the R^2 plane. It is well known that some geometry problems (such as the angle trisection problem) are unsolvable with this toolset.
David Hilbert, who proposed the first formal system of axioms for Euclidean geometry, used a different set of tools. Namely, he used some imaginary tools to transfer both segments and angles on the plane. It is worth noting that in the original Euclidean geometry, these transfers are performed only with the help of a ruler and a compass. Although Hilbert geometry is very similar in its results to Euclidean one, it nevertheless differs significantly from it. A striking feature of the Hilbert system of axioms is the complete absence of circles. For this reason, it is impossible not only to trisect an angle but also to intersect two circles. In other words, it turned out that in Hilbert geometry one can construct even fewer points than in Euclidean geometry. When Hilbert's contemporaries called his attention to this fact, he added to his system of axioms the so-called "completeness axiom", the essence of which is reduced to the following statement: "Let there exist all points of the plane R ^ 2 that I could not construct". In other words, Hilbert cannot construct the intersection point of two circles with his tools, but he can prove that it exists using the axiom of completeness.
The followers of Hilbert circumvent this difficulty by additional principles. For example, Hartshorne directly adds to Hilbert's system of axioms the axiom of the intersection of two circles (E). Now in such an extended geometry "Hilbert + E" it is possible to construct all the points of Euclidean plane. However, the following question arises. Since the addition of the circle-circle intersection axiom (E) expands significantly the constructive capabilities of Hilbert's geometry, then maybe his original tools for transferring segments and angles turn out to be superfluous in this extended geometry? This indeed is the case! However, to eliminate Hilbert's tools, some of his other axioms must also be modified.
There are also other known flaws in Hilbert's geometry. For example, Hilbert's treatment of angles cannot be considered completely satisfactory. The point is that Hilbert only defines interior angles (greater than null angle and less than straight angle). As a result of this truncated definition, it is not possible to universally define the addition of two arbitrary interior angles. The sum of two interior angles in Hilbert theory is determined only iff the resulting angle is also interior. Thus, the addition of angles in Hilbert theory does not form a group. Another drawback is that Hilbert does not give a clear definition of triangle orientation on the plane, although he uses this concept to define polygon areas.
In my recent work
Evgeny V. Ivashkevich, On Constructive-Deductive Method For Plane Euclidean Geometry, arXiv:1903.05175
I abandoned Hilbert's tools and replaced them with the Euclidean ones (ruler and compass). As a result, the constructive part of my development remains faithful to Euclid. At the same time, I have modified some of Hilbert's axioms in such a way as to give a modern definition of order and addition of angles (both convex and reflex) and of triangles orientation on the plane.