In general which laser would cut clear glass more easily, a visible spectrum laser or an infrared laser? Assuming the two lasers have identical characteristics and power output except their wavelength, which one would be more successful and easy in cutting clear glass?
In case there is doubt that you can even cut clear glass with a laser, see here these two video demonstrations:
Visible spectrum blue laser cutting glass
Infrared laser cutting glass
Secondly, what is the refractive index of glass at infrared wavelengths (~1.52 at visible white light)?
 A: In industry, cutting lasers are not selected based on their wavelength.  They are selected based on their cost, which is closely related to energy efficiency.  Carbon dioxide lasers, which are infrared lasers, are used because they have the best energy efficiency.
"Clear" glass will absorb some portion of the infrared spectrum, as will any other solid. So some infrared radiation, depending on the wavelength, will cut the glass better than visible radiation, which is not absorbed.  The exact mechanism of cutting can vary.
It is also important to consider the geometry of the beam of radiation; owing to the diffraction limit, this is not independent of the wavelength.
A: You dont define what it means to cut easier, but I will address how you select wavelength to cut faster and how you can select wavelength to be more precise in your cuts (although this precision likely does not matter for the vast majority of laser cutting applications)
For lasers that differ only in wavelength, you would typically want to select the wavelength with the highest absorption in the material you want to cut, if everything else was equal. Photons that are reflected or transmitted through the material won't transfer their energy to the matetial you are cutting and therefore wont contribute to the cut. This way you maximize cutting speed. Obviously other considerations come into play with practical lasers, with laser power being a big one.
If you want to do extremely high precision cutting (micro/nanoscale), you will tend to want a shorter wavelength. Lasers with a shorter wavelength can be focused to a smaller spot size, increasing precision. This is only important at micro or nanoscale cutting.
There are other ways a laser can cut a material than by direct absorption in the material to cut, but I will ignore those for this answer since you have control of the laser wavelength.
