Greater Flow rate - 5/8" or 1/2"

Assume that water is incompressible. That's a pretty good assumption for this problem!
Neglect the effect of connections. There are the same number of connections in each case anyway, so this won't affect the comparison.
The first two parts are the same in each case (unless I've forgotten how to read or you've fogotten how to type), so given the above two assumptions, we can forget about those, and look only at the effect of the third piece of hose.
I'll also assume that the numbers you gave are inside diameters.
So, we are left deciding if 30' of 5/8" will allow more flow than 10' of 1/2" hose. When I type the numbers into my personal flow calculator, I get *slightly* more flow in the 30' of 5/8. Let's look at why...
In turbulent flow, for reasonably small changes in D, you can consider that the pressure loss for a given flow rate is roughly proportional to L/D^5. (Changes in friction factor means it's not exact, but for small change in D, that won't matter too much.) The proof is left as an exercise for the reader...
So, going from 1/2 to 5/8 is a 1.25x increase in diameter. 1.25^5=3.05 But you've got three times the length, so for the same flow rate you'll have 3/3.05 =0.984x the pressure loss. But you have the same pressure in each case, so you can get a *little bit* more flow out of the longer but fatter hose.
I suspect it's not a coincidence that the answer is so close to "the same"...
-Paul