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For the robot arm part, you can make an autocad/google sketchup/<other drawing="" software=""> drawing which can be exported as an stl or dae file. Use the stl file in a urdf file - and gazebo will be able to visualize it for you.

This urdf tutorial has a section on how to use meshes (which is what you want) at the end. You could also make a custom geometry based arm using bars, and so on - but it'd be a bit more complicated.

You don't need a physical arm to do any of this.

To do the inverse kinematics part, MoveIt is the most popular that I've seen - but I've never used it - so, I can't coment about it much. have a look at it here :