Hi, will try to answer some of your questions:
1. First some background. Since your operation can be achieved by manipulating the work-piece, a stationary laser head will work. Some industrial laser types such a Nd:YAG or Diode work quite well with robotics because they can be delivered via fiber-optics. the CO2 is less flexible from a deliverability stand point but in your case shouldn't be an issue.
2. The Co2 and Nd:Yag have a longish history of being used for cutting, the primary difference being the CO2 generates a continuous wave while the YAG is pulsed. At low pulse rates and high speeds a rough edge can be formed with the YAG but with the right combination of speed and pulse rate, an acceptable edge is achievable. Talk to the OEM and they will be able to guide you, there is tons of literature available. One benefit of the YAG over the CO2 is that its wave length lends itself better to being absorbed by a shiny metal like SS or Al.
3. Both laser types can easily generate the energy required for your cutting operation, however when I worked with lasers about 10 years ago the Co2 was easily the most efficient laser on the market about 10% energy conversion efficiency as opposed to about 1% for the YAG all other things being equal. That's something to keep in mind while talking to your OEM.
4. Since the energy conversion efficiency for lasers are low, they usually come with chillers (waste heat). The part itself does not need cooling per se...the spot sizes at about 10 microns and the focused beam leaves a minimal heat signature. You can shroud the operation with an inert gas like Nitrogen or Argon if you like if you do not want the edge to oxidize, for many commercial applications there generally isn't a need. With the Co2 laser there is a gas mixture which achieves the 'lasing' if you will, so CO2 gas would be a consumable and will need to be replaced, although I have heard there are sealed CO2 lasers available that don't have this requirement.
5. Finally, there are big name OEMs like Rofin/ Coherent or Trumpf. Others like Kern or Vytek are worth a look. I worked with a couple of small OEMs and had a pretty good experience, will pass the names on when my memory comes back a bit.