You will definitely have to clear US immigration and customs with your bags at LAX, even if they are tagged all the way through to HNL. There should be a belt where you can recheck the bags for the UA flight once you leave the INS area. You'll then move landside to take a shuttle to UA. If RG is doing their job, someone will contact you upon arrival and order the STAR Shuttle to take you over to T7, rather than having to use the regular airport shuttles. [This is a benefit for STARGolds flying C/J or F.]

As noted, you are unliketly to be able to use any of the Bradley Terminal lounges as an arriving passenger. Most likely all the STAR-affiliated ones will be closed, since they only open within hours of the departure of TG and SQ flights. MX does not operate its own lounge, and the official STARGold lounge here is the RG one, which is open limited hours. There are no official Arrivals Lounges at LAX for STARGold. If you are flying RG First, they may have arrangements with a local airport hotel.

The RCC will have very limited facilities, and few I have visted in the USA have showers.