Bleed X and Bleed Y: Bleed option extrapolates texture edge colors outside texture mapped area.
Blur: A smoothing filter. The measuring units is the proportion
radius of the smoothing filter/image width
Therefore, quite small values are usually sufficient. Level 0.01 smoothing corresponds one percent of the image width which usually blurs the image quite a lot. Note, that 'Gradient' feature also smoothens the pixel boundaries, but smoothing uses only 2*2 pixel blocks. Blur smoothing is not limited, it can use e.g. 10*10 pixel area or more for blurring.
MipMap: A special antialiasing technique for texture maps. If the value is greater than zero, the program computes a sequence of mipmap images. Each image halves the resolution of the previous one. The last image is only one pixel wide and contains the average color of the whole original image. The render engine uses interpolated colors from mipmaps whose resolution is suitable for the image area being computed. The higher the MipMap value, the more low resolution mipmaps contribute into the result.
The Edit Bindings part of the tab shows the available image channels and the sub channels of the output channel. The existing channel bindings can be selected and then removed using the Remove button. New bindings can be created by selecting an image channel and an output sub channel and by pressing the Add button.
The Animation tab of the Texture object is the same as the Bump object has. See Bump documentation above for details.