In the left side menu is a icon shaped like an eraser. This in effect wipes any graphic or line when you select and the pass the tool over an area (a form of deletion tool)
The eraser, the marker/pen, the Marquee Tool, and the Type Tool.
Photoshop have Horizontal Type Tool and Horizontal Type Mask Tool which you can use to type text in Photoshop.
The function of an eraser tool is to remove the mistake that has been made.
Photoshop is not bad, it is simply a tool.
In Photoshop, the pucker tool is part of the Liquify tool. The pucker tool is used to move pixels towards the center of the brush area.
Photoshop is one such tool for that. Pixlr is another tool which can be used like Photoshop.
All tools like Move tool, Brush tool, Type tool live in Toolbox or tool panel as you say. It is one of most important panels in Photoshop, you can not access tools in Photoshop from any other panel in Photoshop.
Photoshop Lightroom does not have such a tool. Anyway you can use external editing software like Photoshop or Photoshop Elements. See related links for more details.
There is no eyelash tool in Photoshop, but you can create a preset paintbrush so you'll only have to design them once.
The Move Tool allows you to rearrange layers in Photoshop by dragging and dropping.
Erases part s of image based on color range. When you click on part of image this tool will erase all portions of image with similar color levels. This tool is most useful for removing parts of pictures with solid color. See related link.
Its the Zoom tool