I am not an electrician, but maybe my comments can help you find answers to all of your questions.
Some welder manufacturer's instructions are very clear in specifying the breaker size, the wire size and length and extension cord size and length. If you welder’s instructions are not that specific you can use the nameplate info to size the circuit. You need to know the primary (or input) current and the duty cycle for the rated output. The rules for welder circuits are different than those for general purpose branch circuits.
I don’t know of any welders that use a neutral, just hot, hot and ground. So to use this 6/3 with ground you would cap the neutral at both ends and use the two hots and the ground with a 3 wire receptacle. The breaker and receptacle size depends on the welder.
You can use an extension cord with your welder, it just needs to be the proper size and type. Type "SJ" (Service Junior) is rated at 300 volts and has a thinner jacket than type "S" (Service) cords which are rated at 600 volts. If used outdoors it must also have a "W" (Wet location) in the description. Type SJ cords are lighter and more flexible than type S cords but are not recommended for heavy use or abrasive environments. A "T" in the cord type designates a Thermoplastic insulation and jacket, which would be stiffer and harder to work with than rubber. Types SOOW and SJOOW use a rubber insulation and jacket. The cord plug and receptacle you choose depends on the welder’s requirements. Also, be aware that the nomenclature is not consistent when referring to cable, such as Romex, and cords. It common to refer to 3 wire with ground Romex as a 3 wire cable (even though there are 4 wires in it), where a 3 wire cord is just that, a cord with 3 insulated wires.
You do not “lose amperage” with extension cords, but there will be a drop in voltage as the length of the circuit increases. So, unless you plan on doing all of your welding 100’ from the garage you may want cords of different lengths. I’ve heard it recommended that if you have a long cord to uncoil all of it when using it, something to do with induction, phase shifts and arc characteristics.
If your instructions aren’t much help, give us the model and specs of your welder and we will go from there.
Horses don't dance.