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The old quip about the only things certain in life are death and taxes could be added to: and concrete will crack!
Concrete cracks for a variety of reasons, but the most common one is normal and not a durability or structural problem: shrinkage. Water is necessary for concrete to cure (it does NOT dry, it is a chemical reaction called hydration), but we will add more water than is necessary for hydration just to get the concrete to flow so we can work it into the shape needed. As the concrete is curing, excess water is evaporating, and the volume of the concrete in your driveway shrinks. This causes it to pull apart from itself, and that is why you see the small cracks develop shortly after pouring. Under certain conditions they even begin the day of the pour!
Sawing control joints in the slab is an attempt to control where the cracking takes place, so it is not unsightly.
There are other reasons for cracking, but an explanation takes much more time; and from your description I think you are witnessing shrinkage cracking.
What can you do? Nothing at all. It is a normal part of a concrete slab.
Check the frequency and depths of the control joints/sawcuts they installed for your driveway. They all are subject to industry standards. A driveway that is 4" thick should have control joint/sawcut depths of 1" deep (24% of the total thickness) and not more than 12 feet apart. Check these out and if he has installed the concrete within these limits then it is difficult to warrant random cracks that appear.
The biggest issue we run into is not having enough black dirt (quality soil) on on hand. Trucking in more soil can add thousands of dollars, and it's not often evident at the start of construction or remodeling.
You can install a flexible downspout extension on your downspout that is next to your front step and run it behind your bush out past your mulch to keep the water from washing your mulch away. You can buy the flexible downspout extensions at Lowes or Home Depot.
What is the water source? It looks like it is coming from a drain pipe behind a shrub? I would recommend relocating that to dump to a different location. No matter what type of edging you install on the bed with the mulch, a massive amount of water from that down pipe will cause some overflow and mess. Maybe you can incorporate a stone trail where the water flows, if you can't relocate the down pipe. Ideally, I'd want to see the drain pipe go under the side walk, and out by the grass, or if its by a garage, down the driveway.
Using warm soapy water is the best for daily cleaning. There are many granite cleaners on the market that will help you if you really want to scrub and bring back that original shine. Laticrete (formerly DuPont) stone care products are great. They make a revitilizer spray that is excellent for weekly cleaning. It has a bit of sealer in it as well, so it will help keep those tops shiny.
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