I can't speak to the current arrangement with United. In the past, there was an interline agreement with Winair. In order to be eligible for baggage to be shipped from your US origin airport and all of the way to SBH you had to buy both your mainline UA ticket and your inter-island WM ticket on the same itinerary. Using St Barth Commuter as your inter-island airline was not an option, and many found that the cost of the Winair ticket purchased through UA was significantly higher than purchasing it direct from Winair. If you do book both UA and WM on the same itinerary then the tagging of your bags all of the way to SBH should happen automatically. Check the bag tags to be sure.
Re: Tradewind, sure they do, with UA, AA, and BA. Same rules, both Tradewind and your mainline tickets must be purchased on the same itinerary. Same caveat, there may be a price difference.
When returning to the US, no matter how your bags are tagged, you must reclaim them prior to Customs, and recheck them after Customs. Some airlines at some airports have a convenient bag check as you exit the Customs area. Not all airlines, and not all airports do. (SJU, for example, has none in its current configuration, which involves clearing Customs in Terminal A). If there is no convenient bag check then you must take your bags to your airline's checkin counter.
Everyone returning to the US with an ongoing connecting flight must also re-clear TSA security before they can proceed to their gate.