First, what you have done with the inner cable at the carburetor end appears to be correct. It seems to fit properly into the hole in the anchor fitting.
Second, those fuel lines look like they have gone hard and are discoloured. One has a strange kink as if it has been overheated. You can get fuel hose at a mower repairer, but there may be cheaper options if you measure the hose and then buy a metre of it.
At this point I have no idea how the upper end of your throttle system is supposed to work - I can't tell from those pictures. Let's see if someone else can help.
