The Curt Trailer Hitch Receiver - Custom Fit - Class I - 1-1/4" # C11567 you were looking at for your 2020 Toyota Corolla Hatchback SE has a 200lb tongue weight rating and a 2000lb trailer weight rating so in that regards the hitch will not have any problems with your 589lb trailer.
Since Toyota and the trailer manufacturer told you there would be no issue, and the hitch has the correct ratings for the trailer I cannot see any problems you would run into.
Regarding sway, that comes down... view full answer...
I went ahead and looked up your owner's manual for a 2022 Toyota Corolla Hatchback XSE, and the manual says that all transmission types of your vehicle cannot be flat-towed with all four wheels on the ground.
With that being said, you can tow your vehicle with the front wheels raised by using a tow dolly. For this route, I recommend the Demco Kar Kaddy SS Tow Dolly with Disc Brakes - Foldable - 4,800 lbs # DM9713045. With the Kar Kaddy SS, you do not need to go through the pain of installing... view full answer...
Depending on what roof rack you have on your 2002 VW Beetle there may be some components that would transfer to a new vehicle but as far as the complete rack transferring over that would be very unlikely.
I'd need to know exactly what roof rack you have on your Beetle as well as the year of the Corolla you have. view full answer...
For hitch balls we recommend white grease like the part # 58117 since the ball moves around in the coupler but for the hitch itself there isn't a grease recommended for that. view full answer...
For lighter duty vehicles like your 2022 Toyota Corolla Hatchback there are only 1-1/4 inch hitches available like the Curt # C11567 as it's only the heavier duty vehicles that have the 2 inch hitch size. view full answer...
When using any extender or adapter like this one # HT5R you will loose 50% of your hitch's capacity. This hitch # C11265 will fit your 2019 Toyota Corolla. I recommend getting an aluminum carrier like this one # C18100 because it's only 29 lbs so you can still get 71lbs of cargo loaded up. view full answer...