1. When Direct Selection Tool is active, Cmd-click it on a path to switch to Path Selection Tool with which you can drag the entire path or select the path for a transform. Another Cmd-click will switch back to Direct SelectionTool.
2. Maybe you are you drawing shape paths inside an already existing Shape layer which has the Layer Style, or creating new Shape layers inside a Group layer which has the unwanted Layer Style.
3. If you enable "Snap Vector Tools and Transforms to Pixel Grid" in Preferences > General, then orthogonal lines will not be anti-aliased. For general lines and curves without anti-aliasing, I think you'll need to stroke paths onto pixel layers with the Pencil Tool.
1. With smaller shapes that are difficult to select, try selecting their layer first. If you are not sure which layer, switch to the Move tool (V), hold Ctrl and click on the small shape. With the default behavior in Photoshop you should be able to select the layer and then use the Direct selection tool to select points on the vector path of the small shape. If you want to just move the entire shape when its the only path on a layer, aside form using the Path Selection tool (black arrow), you can just use the Move tool (V) to change the position of the layer. If you have to deal with a lot of shape layers that you just want to click and move, enable the Auto Select for the Move tool in the Properties bar. Then holding Ctrl while using the tool will toggle to its non - Auto Select mode.
2.On the Tool's property bar under the main menu, make sure the chain button is not pressed then click on the Style swatch or arrow to the right of the chain button to open the style swatch popup and from its menu there choose No Style. Next shape layer you make will not have the current style automatically applied.
3. I'm using CS5 on a pc and creating 1 px straight line with the Line tool is buggy with its alignment to grid. The Line Tool draws lines in Shape mode as rectangle vector paths. The difference between the Rectangle Tool and the Line tool is that the Line tool aside from having a Weight attribute, it draws the rectangles starting from the middle if its side. In order to snap it to grid you have to set the grid in the settings to Gridline every 1 pix and Subdivisions 2. If properly working this is supposed to solve your problem if you enable snap to grid and draw your lines starting in the middle of a pixel. The problem is that this doesn't work properly with the Line Tool which does not snap perfectly to the grid. The work around is to draw your lines with the Line Tool in Fill Pixels mode which has an Anti-aliasing option that you can turn off.
Another alternative for drawing lines on raster layers without anti-aliasing is to use the Pencil tool and hold Shift while dragging with it. The Pencil Tool in Photoshop is the same as the Brush tool but without anti-aliasing. I think more proper name would be non - anti-alised brush.