It sounds like you are describing garter stitch. That is, if you are knitting back and forth and not "in the round". If you drop a stitch and need to pick it up, you will need to pick it up as if you did a row of knit and a row of purl. I hope that makes sense. If you are only knitting, when looking at one side you are actually looking at a knit row and a purl row, because on the wrong side a knit stitch is a purl. A crochet hook would make it easier. You would pull a ladder through a stitch below it as if it were a knit, then pull the next ladder through the stitch as if it were a purl.

A photo would help, but have you tried pulling the stitch through in the opposite direction, front to back or back to front? Or turn the piece around and work from the other side for that row if that's easier for you.

That's another way of saying the same thing -- a knit stitch has to be pulled through as you would knitting and a purl as if purled. The previous row loop goes in front on a purl row and the back of a knit row. When you turn and knit the next row, you get the smooth face of the knit stitch on top of the bumpy loop that was the back of the last row. If you reverse the direction you pull the dropped loop through, I think that will fix that for you. And a crochet hook makes it much easier for me.

The splitting is another issue. Been there. I just returned some yarn I bought to make a shawl. I read comments on Ravelry about it splitting terribly after I got home and went online to find the pattern. Looked at the yarn and realized there was almost no twist to the strands. Wasn't worth it.