Is there any specific criteria (temperature, material, etc.) to use coated carbon steel bolts instead of alloy steel bolts in stainless steel piping or machine anchoring?
Temperature for sure. For flange bolting, B7 studs and 2H nuts (low Cr-Mo alloy) are used more or less as standard to about 800 F, and may be permissible for use to up to 1000 F under certain circumstances. Only above 800F, or in services where EXTERNAL corrosion is a severe concern, are alloy studs/nuts typically specified.
Coatings are a matter of preference, with some using fluoropolymer coated fasteners exclusively, while others use black oxide or zinc plated fasteners, depending on external corrosion conditions.
For bolting stainless equipment to the structure, construction bolts are typically used, and these are hot dip galvanized.