The towing specs are in the handbook. My mk.3 2018 cvt is OK for towing, don't know about your model.
Unbraked trailer weight is dependent on the vehicle weight but it is likely to be in the region of 450kg, you need to check the manual. Braked trailer weight will be higher, maybe 750kg or something, but again check to be sure.
I bought a Witter towbar for my Yaris from the Towing Centre back in 2021, I chose the detachable Swan neck just so I don't keep banging my shin on the ball when putting shopping in the car (done that!).
Brink make towbars which are often sold as vehicle manufacturers accessories. Westfalia are another good brand.
Any towbar fitted in the UK must be type approved and marked.
As for the wiring, it depends on the vehicle and who is doing it. I used a generic kit but it takes longer so if you're paying for it to be done then it might not save anything.
As you say, 7 pin for regular trailers, 13 pin for caravans with battery charging etc.