Electronic – No components 3D view in KiCAD

You can rotate components while keeping wires connected by clicking g for grab, then clicking r for rotate. It's very convenient.

But like you noted, it doesn't keep any right angles you chose, and instead connects them with the shortest route from the closest break-point in the wire. By breakpoint I mean some point that isn't a straight line.

I don't know if there's a way to force the wires to behave how you ask. There might be a setting to forbid any angled lines in the schematic. And then that setting might behave how you want when you rotate. But you can imagine it's hard to keep wires "neat" when you go about rotating them from the initial configuration. It would require the program to dynamically organize the wires itself.

I generally want signals to flow from left to right, so I put inputs on the left of the box, and outputs on the right. Power connections (if shown) go on top, and power (again, if shown) on the bottom edge. The box must, of course, be tall enough to show all the pins, but only needs to be wide enought that the pin names don't overlap. I would usually put the reference designator and part type above the part, not inside it.

Spacing of pin/signal groups, as in your right example, can be helpful in separating functions on a part.