Getting the drainage to move water away from the house foundation is probably the most important issue within your question, I would let that be your guide as to whether the concrete stays or goes. Don't take somebodies quick answer that 'things will be ok', get a level, sit down on the concrete and scope things out for yourself. Can asphalt be laid over concrete? For sure! Concrete highways are routinely resurfaced with an asphalt topping.

The walkway needs to be removed. There isn't enough space to add a layer of asphalt over the concrete. See where the door threshold sits just a smidge above the concrete? And the glass block windows?

ETA: The concrete walkway is a bit of a barrier to cut down on tracking black stuff into the house when the asphalt is freshly laid.

I'm year 15 or so into a new concrete driveway. We paid to have really top quality concrete, busted our behinds to not use salt on it for x period of time, and made sure to shovel snow-salt infused crap dropping off our cars into the driveway . We also didn't drive on it for xx period of time when it was first poured and sealed it with some sort of sealer. My driveway didn't have that sort of pitting when new.

My driveway doesn't have the pronouced horizontal lines across it that your driveway shows. Perhaps there is someone on here that can give you an expert opinion on the quality of the mix that was used.

My guess (and that's all it is) is that he concrete was poured on a hot day and/or no curing agent was applied to retain near-surface moisture during curing. Unfortunately, you're stuck with it as it is now.