Why is water leaking behind my new rain gutter?
The problem is partly that your drip edge isn't dripping. It may have been pulled too tight when it was installed, causing it to lay too flat. It's not really a gutter problem. The fact that rainwater was running down the fascia was an issue that needed resolution anyway.

You can try to adjust your drip edge by flexing it downward, but it may be better to fit some flashing behind it that extends into the gutter. You can purchase suitable galvanized steel or aluminum from a home supply store and cut it to fit. 

The downspout solution seems ok to me. The only bends available are much too large for that situation. It shouldn't be a problem at all unless there's a blockage.