I can see why you are pining to get rid of the southern driveway and SW garage. Your garage is in the Southwest and I'm assuming this is a source of shade on your small garden to the west of your house, and that your house is also a source of shade in the latter part of the day. Is that right? Bummer. And your driveway is on the South... double bummer. Yeah... I would definitely consider getting rid of those. Perhaps you can move your garage to the South East Corner?
To the south of the house is the driveway (which takes up almost all of the property on that side), and to the southwest is the garage. Eventually I'd like to remove the garage and driveway in favor of a larger garden, but that is far away and may never happen.
When you build your trench/path system, you really should use perforated drain pipe. Use round stones (called drain rock), not crushed angular gravel, and put that around and on top of the pipe, and do put landscape fabric on top of the rock to keep silt/dirt out of the system, then, put your path paving stones. This gravel/pipe/landscape fabric will improve drainage greatly. Here's a blurry but sufficient youtube video of a guy doing it. Skip the first minute and watch until minute 4ish.
fill the trench with gravel (what kind? will gravel work? do I need landscape cloth or a pipe or anything else?)