Take the swan neck off the downpipe and put straight into gulley also change you're guttering to deep flow and lift in up it's too low trays will help stop water going behind gutter but won't stop the splash over
This
https://www.dortechdirect.co.uk/catalog/product/view/id/2327.html?gclid=CjwKCAiAjs2bBhACEiwALTBWZbCqWegEy7lqbg9frgZ1cw-3AclsGRlWGJxBUxJ4Hb7n_gBmresbzRoC_l0QAvD_BwE
As above do gable same pitch or you with end with slithers for tile towards the bottom can't you offset the extension or do same pitch and hip end the extension
There is a fair few solid core 35mm primed doors doors not solid wood on the market now most DIY outlets stock them but there prices do vary £60upwards
How long is the section to the right of the pic if not to long replace the lot or remove the fake oak piece cut down till the two rounded sections meet