A Craig - your roof is what is known as a Mansard style. Search 'Mansard Style roofs' on Houzz for a lot of examples that may give you some answers (see one below). Obviously this home was built pre aluminium guttering that does not rust so to replace the existing guttering with seamless aluminium should solve your problems. It would be worthwhile replacing the downpipes with aluminium too. A bigger problem you could probably have is matching the existing tiles, but from photos on Houzz it does appear you would be able to get away with a different tile or even use a slate were you to change the flat area to a slightly pitched roof. Your existing tiles can be resprayed a different colour too. Call in the experts, the more people you speak to the more ideas you will get. Afraid whatever you do will be expensive, but consider it money well spent to alleviate any future water problems - water does so much damage that will negate all other renovation work you do in the house. [houzz=]

Roof Gabel vs pitched

your roof appears to have various angles and shapes with eaves for some parts and not others and the flat roof at the rear could be a wasted opportunity for an upstairs deck and this may be intentional but i suggest that the best roof would be determined by orientation and climate..a copy of the floor plan showing direction of north would help understand the roof better

you'll have a much better BASIX energy savings rating and a more comfortable home with a light coloured roof and the house will look so much better if you consider white walls, black window frames and one of the light greys like Windspray or shale Grey for the roof with gutters in black, fascias black or white, downpipes matching the walls and garage and front doors in Timber tones...enjoy your new home