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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.
Tools are essential to work in Photoshop, there are Move tool to move objects, Type tool to type text, magic wand to create selections, zoom tool to zoom image in and out and many, many more.
Go on http://www.photoshoplab.com/photoshop-tool-basics.html and it tells you all the tools meanings (:
Selections tools inside Adobe Photoshop are : Lasso tools, Marquee tools, Magic Wand and Quick Selection, Color Range from Select menu and you can also trace edges with Pen Tool or Magnetic Pen Tool which is accessible when Freeform Pen Tool is active.
There are around 100 tools and features on Photoshop, and you will just keep finding more and more :)
history brush tool.
The eraser, the marker/pen, the Marquee Tool, and the Type Tool.
It is question for Photoshop and Illustrator developers, which tool is from which app but Illustrator have Photoshop raster filters, that's for sure.
Lasso Tool for selections, Pen Tool, Pencil Tool, Brush for drawings and painting.
Selection tools in Photoshop: Marquee Tools (M), Lasso Tools (L), Magic Wand and Quick selection (W). You can also select color from image as foreground or background color with Eyedropper Tool (I). Navigate with mouse above any tool in Photoshop and you will see tooltip with keyboard shortcut for that tool in parentheses.
There isn't drag tool in any Photoshop version. To move things around you can use Move Tool or Path and Direct Selection tools to move paths and shapes.
Brush is on left side of window at so called tools palete, it is 8. tool from top to bottom. With this tool you can paint in Photoshop. See related link.