I'm not sure all the details of the Solar System formation are understood, but the general principles are well established. The dust cloud from which the Solar System formed was probably roughly homogenous. However once the Sun began to form, the dust cloud around it rapidly became differentiated. The heavier non-volatile elements stayed near the Sun while the lighter more volatile elements were blown outwards. That's why the inner planets are rocky while the outer planets are gaseous or icy. Incidentally, the water on Earth probably came from comets after Earth was formed, though views are mixed on this subject.
The dust cloud was probably formed by supernovae of Population II stars, and you're correct that these would have left remnants of some form. However dust clouds are big, and were even bigger when first formed. The nebulae where we see star formation have had time to contract and become more dense since the supernovae that formed them. In addition, it's been (at least) 4.5 billion years since the last nearby supernova. There has been plenty of time for the supernovae remnants to wander around, so it's no surprise we don't have one next door.
Having said that, I've never seen an estimate for the density of Population II supernovae remnants, and I'd be interested to know if there's a firm number for this.