I occasionally get surprised by a ball that seems to be hitting the net but somehow just clears it - I think quite a few people do. As already said, the best tip is to get there and cover it even if you're 99% sure that it's in the net. I would imagine that the more time you spend at the net the less you will lose track of the ball because following it over the net will become more natural to you.
Both of the shots you describe are pretty horrible to have to hit a volley off but there's probably one thing you can do to make them more tolerable - be fast. When you split, make sure you're on your toes and ready to move. Stay focussed at the net and ready to react to anything the ball will do. The first scenario is very coach because the ball could do an enormous variety of things after hitting the tape, it might pop up and let you kill it, just barely dribble over the net and leave you picking it up off the floor or possibly kick over your racket. A slight direction change of the net cord is something else you may have to deal with. Basically, the only thing you can do it make sure you move well and stay light on your feet. If this does happen deal with it as best you can but there isn't any unique technique to the shot, good footwork definitely helps.
The second one is another tough shot to deal with. Trying to put it away or even just hitting a quick volley off this is very difficult and usually gets put long/wide. You can't just hit a flat, soft volley or a drop volley because the pace coming in will most probably make it sit up and get crushed. The higher you can take the ball, the better because picking up a fast shot from below the net is even more difficult. The underspin suggestion is something I would also recommend, it allows you to keep the volley in play a bit easier and makes it more difficult for your opponent to pass you with the next shot. If you can get to the ball and take it at a decent height above the net, you can often punch it away using the pace of the incoming shot but this is very rarely possible on the shots that you describe - they simply don't clear the net by a big enough distance.