NEC Sec. 430.122(A), circuit conductors for an adjustable-speed drive system must have an ampacity of not less than 125% of the rated input current to the power conversion equipment. I am not aware of any specifications as to type of use. My understanding is it has more to do with the maximum duty capabilities of the VFD, and too safe guard the wiring should the motor be changed to something different. It also has to do with providing sufficient headroom when the VFD is run in an overload scenario which can be up to 200% or more of rated output for short periods of time. Not an electrician, just what I have read and seems to be continually referenced. There is also local code which may have other requirements.
Answering your NEC questions.
www.ecmweb.com
I would refer back the the lathe manufacturer as to the required wiring and recommended single phase breaker size for the power to the lathe. I will say, that a single phase 30A is not sufficient for this size VFD/motor. I have done a number of VFD systems with Yaskawa drives and for a 5 Hp motor, they usually they are run off of a single phase 50A breaker. The input current for the Yaskawa 0030 model is 24A at 240VAC (3 phase), on 3 phase it would require a 30A three pole breaker, on single phase would be ~40A x 125% = 50A. I have not seen how one address it being run in derated mode, breaker fusing is also affected by if you are using a DC choke (DCR) which decreases the current pulses, so they will lower the fusing breaker requirements when used. Example below is from the Fuji manual.
View attachment 443687
In addition if you do use a receptacle, current NEC code states GFCI protection is required for all 125-volt to 250-volt receptacles rated 50 amps or less, not exceeding 150 volts to ground, supplied by a single-phase branch circuit in the following:
Basements, all areas, including unfinished portions
Bathrooms, all areas
Crawlspaces, all areas
Garages, including service bays and similar areas, but not show rooms and exhibition halls
Indoor wet locations, all areas
Kitchens, all areas, as well as other rooms for cooking or food or beverage preparation, such as break rooms
Locker rooms, all areas with showers
Outdoors, all areas
All areas with sinks. Not required for industrial laboratory sinks where the removal of power would introduce a greater hazard.
GFCI protection is required for all areas of rooftops of non-dwelling units. GFCI receptacles must be installed within 25 feet of heating, air conditioning and refrigeration equipment and readily accessible from the rooftop’s surface area.
I would hardwire the lathe so you do not need to get an expensive GFCI Breaker, and they tend to not play nice with VFD's.