Bury Downspouts Recommendations

[stryker2000]

I'm trying to determine the best route to bury the gutters around this house. The sunroom has 3 around it and the ground gets saturated, which is the top left room in the picture. The bottom I have out near the sidewalk (orange line) but am debating just running all of them out towards the back since it's a small patch of brush with a street drain to the left (purple). There is also a downspout that I have coming out onto the driveway (bottom left picture). It would need to be routed under the sidewalk and out in the yard.

Can these be tied together as I have shown with pvc? I see some videos showing corrugated. Also, what are the best options in-yard for termination? I see popups around the neighborhood and they are just making puddles.