Re: gutter problem
Not sure I follow your roof line descriptions.
Option 1 sounds like the existing roof line and the new addition line are parallel, upper roof draining to lower roof. This is the optimal and if it were me I'd just extend the drain line down to the corner of the addition rather than snake the downspout as you've described, because you're right, that will be a leaking and clogging nightmare.
Option two sounds like you want to raise the outside wall so that you create a valley between the new and the old. If this is the case, then you definitely DO NOT want to do it this way. You are designing a leaking system into your house, and you do not want to do that.
The other way I'm reading this is that the addition is perpendicular to the existing gutter line and your option #2 is to slope the roof in the same direction as the existing roof so that the gutter is merely extended to accommodate the new roof. There would be nothing wrong with this scenario as it's done all the time. Downspouts do not need to be at the end of a gutter line to function properly. Whether or not it will withstand the load of the existing and new gutters demands will depend on the size of the downspout, the length of the gutter, and your typical rainfall amounts.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!