When I opened a project that was left for a couple weeks, I was stuck in the situation as shown in the picture, where I could not select any other vertices or edges other than the edge highlighted in pink. Even when I tried to click on the other components, it will somehow drag the pink edge and extrude weird faces. How can I solve the problem? Do I need to exit a certain "mode" to get rid of this?
In the screenshot I can see you have all of your object geometry selected, not just one edge: how?
Look at these numbers in the bottom right edge of the window:
they say that you have 251 vertices selected out of the total 251, and so on for edges and faces.
Your pink edge should be an edge with some "special" feature, likely a crease, as others said.
The pink edge is likely an edge crease. Try selecting all (A), then removing the edge crease by pressing Shift+E, keying in '-1' and hitting Enter.
What could cause the selection problem isn't certain. From the screenshot I can see there's the orange selection highlight, so it seems you're able to select faces. It looks like edges are not visible though, have you perhaps turned those off by accident?
You can check this in the 'Overlays' menu in the viewport toolbar(see attached image). Make sure 'Edges' is enabled.
Considering that the X-axis is visible through the mesh, have you checked if the face normals are pointing the right way?
Also, have you tried inserting a new mesh and editing that? Does selecting a different tool help?