I have a similar trunnion style hitch. I agree with all of the above. I generally take the hitch out of the receiver because we use the bed of the truck for things like picking up fire wood and it just gets in the way. Do what works for you.
With sway control, it should not be used to fix a sway issue. If you have a sway issue then you need to look at tongue weight and how the trailer is sitting (too high in front). If everything is setup properly, in theory you can leave the sway bar off and it should track perfectly. Sway control is like insurance, you don't need it until you need it. It will most certainly help when the unexpected happens. High side winds, fast semis blowing by you, evasive maneuvers, can all cause dangerous sway.
Get to know how to manually operate your trailer brake, because even with the sway control the manual lever when pressure is gently applied can help bring sway under control also.
2013 Jayco Jayfeather X20 E, 2008 GMC Sierra 1500 Crew Cab Z71 SLE