Yes, normally a dry well is the termination point for a french drain.
A dry well is normally an underground 'enclosure' built of stone to hold water for reabsorption into the soil. Technically, a dry well is any 'space' in the ground whose depth is greater than its width (yech!). You can do what you state as long as you're able to accommodate the water in a worst-case scenario. As for the landscaping around the house, you can french-drain it, and get it to the dry well. Plus, as you mention, the mulch around the foundation is actually harboring water if there's no slope. Again, you should only need 1/4" per foot of slope to drain water. This is assuming you'd have grass around the foundation; not mulch. If I was you, I'd see if something can be done about the slopping and remove that mulch. AT worst, put in a french drain near the foundation and get the water to the drywell.