Right mouse click over the selected icon on the left side menu bar. This will bring up options for each of the tools in that icon section area. Left mouse click on the one you wish to make active
Photoshop have Horizontal Type Tool and Horizontal Type Mask Tool which you can use to type text in Photoshop.
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.
This tool gives you a sum of pixels you can change the colours of.... So anything you imagine can be done, there are no limits.
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.
Its the Zoom tool
You can do that with Sponge Tool (probably hidden tool at you screen, it is in group with Dodge and Burn Tool), take a look at Options Bar and select Saturate.
It is question for Photoshop and Illustrator developers, which tool is from which app but Illustrator have Photoshop raster filters, that's for sure.