The patio surface should be sloped away from the house. If they are the interlocking concrete pavers (4"x8" or so, depending on the shape/configuration), they should be set on a 1" uncompacted sand setting bed. The compacted base should also be sloped parallel to the paver top slope to maintain the 1" setting bed. - If you using the larger "stepping stones" (12x12 or larger), that is a totally different.
I don't see exactly how the second photo relates to the first, but there certainly has to be something done for your flimsy downspout set-up. - Perhaps pvc (not the cheap corrugated stuff) that could be incorporated under the patio surface.
No idea of the size of the patio or what the climate is, so anything else would be more of a guess than my small suggestion above.
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