I'm a beginner, too, and experienced some of the same problems.
Re the sheet: it may have some kind of sizing in it that makes it slippery; also, I read somewhere that sheets are hard to hand-quilt.
Re the slipping ruler: you can put little circles of extra-fine sandpaper on the backside to help hold it in place. There's also a plastic called "InvisiGrip" that can be put on the back of the ruler: JoAnn's has it, I'm sure other places do also.
Also, slow down, cut a little, stop, move your other hand ahead, then cut a little more, etc. I find that the ruler seldom slips right where my "holding" hand is, so I've been moving it more often to avoid the slipping.
Don't give up! It takes time to learn how to cut accurately, but it saves time if your pieces are "right" to start with.
Jory