Issues with the Free Transform Tool

Another issue is that when creating a square shape, it always starts with 1px rounded corners, which should not be the default.

With the Rectangle Tool and no Shape Layer selected try setting the Radius to 0. 

Another issue is that when creating a square shape, it always starts with 1px rounded corners, which should not be the default.

By @Simple Brand Studio

I cannot reproduce this on my end. In fact, even when I change the corner radius to 1px on each side after drawing the rectangle, creating a new rectangle results in 0px rounded corners. Which version of Photoshop are you using?

Additionally, the general transform behavior only works in legacy mode, even after updating the settings in Preferences.

What do you mean with »general transform behavior«? 

I am not fully sure what your problem is, Pathpoints do not need to align with the pixel grid per se. 

Even if you checked »Snap Vector Tools and Transforms to Pixel Grid« a transformation can result in a »mismatch« and you might be better of dragging the Pathpoints to »snap«.