All extension cords are stranded. The only reason for 4 wire is when you run either 3 phase or 120/240V in combination. The rule of thumb for circuits is 14 gauge is protected with a 15 amp breaker and should not be loaded continuously with more than 80% of that rating or 1440 watts or 12 amps. For a 20 amp circuit its 1920 watts or 16 amps. That's where you need the NEC code book. Welders are wired based on duty cycle. Its more complicated than I want to explain here*. A bare bones minimum will be a 20 amp breaker for any welder running on 120V. I trip my old Millermatic 90 all the time on a 15 amp breaker. If I run it to the highest settings, I trip a 20 amp easily as well. I have wired it to a 30 amp 120V with #10 and the problems have subsided. The welder protects itself with a thermal breaker, the wire is protected by the breaker. How long the run of wire is from the main panel is also critical. Stick a volt meter on the outlet and have someone look at it while you weld. Look at that voltage which will be less than the static voltage (no load). If you have 120 volts static and you read a voltage below about 110V or more while welding, your developing a substantial voltage drop. The wire is the primary reason for this for this drop given the current welding configuration. Solution, up the wire size/ breaker trip current, increase the voltage form 120V to 240V (or more) or decrease the current draw. Since most welders use a pre-determinded voltage (some are bi-voltage), your only sloution is to increase the wire and breaker size. Most homes (includes rentals) use convenince outlets with 15 amp/14 gauge wire. In my opinion, garages should have 20 amp outlets.
*A good example of wire size not matching breaker size is with AC units. Most homes have a much larger fused disconnect than the wire could continuosly handle. The reason for the large breaker is to handle the extremely large startup current the compressor requires. Once running, the compressor draws significantly less current.
Look on the compressor unit. You will see "maximum fuse size" and "minimum circuit size". Minimum circuit size refers to the wire size. You can always go larger.