ADI is part of Sony's auto-TTL flash system, rather than a separate mode (I think) - 'auto distance integration' or something like that, helps to make the flash exposure more reliable.
HSS is High Speed Sync which allows you to use flash at shutter speeds above the normal x-sync max - 1/160sec on the a450 - which is a separate issue. Another guess here, but with some cameras you can leave HSS switched on permanently and the camera enables it automatically when the shutter speed needs to go above x-sync, and then automatically disables it when not needed.
I would use the full auto mode, with pre-flash, which is generally the best option by far. If necessary, adjust the exposure level by checking the LCD/histogram/blinkies and moderate with the compensation controls. See handbook - I don't know how Sony does this and they're all a bit different with flash management.