Geometry Construction Problems
A good book to consult is "One Hundred Great Problems of Elementary Mathematics: Their History and Solution" (NY: Dover Publications). This is an English-language reprint of a German-language original "Triumph der Mathematik" by Heinrich Dörrie (Leipzig).
It has been proven that any construction attainable with unmarked straightedge and compass can be accomplished with compass alone. Try it on some of your already-known constructions. It should be noted that in this mode a straight line is deemed to be known/constructed if two of its points are known/constructed. Section 33 of the source cited above.
It has been proven that any construction attainable with unmarked straightedge and compass can be accomplished with straightedge alone, provided that a fixed circle (with center) is present in the vicinity. Try it on some of your already-known constructions. Section 34 of the source cited above.
There are constructions which cannot be done with unmarked straightedge and compass that can be done if one is allowed to make two marks on the straightedge (for the purpose of sliding a fixed distance). These are called "neusis" constructions. Try a few of them, e.g., Trisection of an angle; construction of a cube root, construction of angles and regular polygons not constructible by ordinary means, etc.
Try this site, it has 40 different challenges. It might be useful. https://sciencevsmagic.net/geo/
The constructions in your list are the basic (but foundational) ones. Try more complicated variations:
- triangle problems: given 2 angles and 1 side, construct the triangle; given 1 angle and 2 sides; given an appropriate combination of internal lines (altitude, median, angle bisector, ...) and so on (some are more difficult than others);
- "CAD style" constructions: tangents to a given circle from a given point; tangent to two given circles, line tangent to a given circle and perpendicular to a given line, ...
- what arithmetic operations can you construct? (given numbers $a$, $b$, $c$, ... as lengths or segments);
- (points on) curves: parabola, hyperbola, ellipse, catenary, ...
- geometric figures given the area and some other property(ies);
- "minimal" figures: a figure [or one of them] from a certain family that minimises a certain property.
You can also check out all the questions in the "Related" section of this site. :-)