I too thought it was the cap. Pulled it out and was about to go get another when I recalled the cap was only a year old. So I pulled the cap out of my other a/c unit and put it in the non working unit and it still did not start. Then I started to trace circuits and tested each leg of the contactor to ground and saw 120v on both legs, so I thought it was fine and kept looking for another reason. Finally it dawned on me I never tested between the two legs for 240v and sure enough no 240v. I had been seeing backfeed on one of the legs. Turned out the issue was a burned out contact point on the contactor. Replaced the contactor for $19 and all is fine again.

"Keep in mind, his 6 AWG feeding the shed is protected by a 100A breaker." While that is likely incorrect conductor sizing, it will not have any affect (beyond voltage drop at start-up). The problem with long runs to induction motors is one of those 'try it' type things. Depending on how often the motor is started you MAY need to increase the breaker size (and the NEC allows this). Starting lower is at least marginally safer, but it can result in having to replace the breaker, or even replace it if the use of the motor changes). Motor circuits are one of the things that the general rules about wire size an d breaker capacity no longer apply. The OP has the correct idea that the motor protects itself against overload with built in thermal protection and thus cannot overload the feed, and all the breaker is doing is protecting the feed conductors against short circuit damage. 125% of full load motor current for the wires, and up to 350% of the full load current for the breaker. So yes, you can see breakers significantly larger than what is typical for general use branch circuits based solely on wire size.

The installation instructions should tell you what size wire to use. As for the # of conductors, AFAIK, in most cases no 120 volts is required, so you probably won't need a neutral. But again check the instructions. If they specify a 240 volt circuit, you don't need a neutral; if they say 120/240 volt circuit, you need a neutral. HTH.

I don't know that I would say that Builder's grade is an unofficial term. Many different manufacturer's use the term although many manufacturer's model numbers do not go to the extent that Trane does in listing this nomenclature within the model # of the unit. (I know this because of decades of being in this industry. This doesn't mean if you buy a new home that it won't contain a non-builder grade unit.... it depends greatly who is making these kinds of decisions.) Quality wise builder's grade is very low. In some cases even the compressors have cheaper parts and or where they were made is likely 3rd world like China. Since Ingersol Rand bought Trane Company many years back the only thing I really know for sure -- the units are only assembled in USA. With that said, Trane has a compressor manufacturing facility in Mexico and the China location is rumor. Given that Ingersol Rand is international company, the China location very well could be fact. Given the low end nature your AC condenser, I would run it and not worry about rust. It simply would not be worth the risk to mess with it. Typically major name brands use the nomenclature of Builder Grade. Many times the major names want you to buy one of their branded brands when the builder grade fails. It's merely a form of brand recognition to keep selling products. Off brands or brands that you rarely hear about are often considered value brands, some not all in my opinion are better than the builder graded brands. These brands aren't called builder grade for obvious reason... builders don't use them. This is where builder grade name comes from. Builders move a lot of units (complete systems), this is another piece of the puzzle that major name brands want... they are only here to sell systems. So to get the accurate picture you have to take a step back as to why it is the way it is. There are only 7 HVAC manufacturer's in USA a non major brand name OR off brand name unit... where do you think it comes from? Hint: The same line the major name brand comes from... in some cases the brand decal mounted to the unit is the only difference. I can tell you Trane / American Standard is odd ball out, simply because they manufacture their own compressors. Nearly 90% of USA condensers use Copeland Scroll, manufactured by Emerson. Which should tell you if the unit has a Copeland Scroll in it, there is very little difference from one brand to another. Put another way to help you understand: A builder is building you a home, he says I'm going to install a 'whiz bang - HVAC brand'. You say: I've never heard that brand before, and so there's concern and a lot of fuss the contractor will be dealing with. Now that same builder says: I am going to put in 'Trane' brand HVAC system for your new home. Ok, wow that's a name I know. He doesn't tell you it's builder grade unit. It's still a Trane brand. But he doesn't have the backlash of having to ease your concerns of a brand you never heard of. The manufacture gets even more brand awareness from home owner. Trane sells the builder grade unit for a $100 or $200 cheaper per system to builder. Builder builds 1,000 homes --- the more money builder cuts from materials the more the builder makes in profit. If he saves $200 per unit --- 1000 homes --- $200,000 simple math.