If you cut off the sleeeves for leg warmers the raglan part will be too wide anyhow. Use the excess sleeve fabric to put in the washing machine and see how much it shrinks.

Just throw the sweater into the washing machine with a warm wash and put it the dryer on hot and it wlll felt. There is no way to tell how much it will felt until it does.

If the skirt is a woven you wont get the same results . I doubt that you will get any felting from the skirt. It might be a worsted woven and that is not going to felt, especially with the non wool fibers. But just throw it in with the sweater and see what happens. It will shrink, but not really felt.

What kind of lining for what kind of project? Is it something where the lining has to be a close fit to the outer shell? Something that will be washed after it's constructed? A hot washed felted sweater can be cozy, but as mentioned it can be unpredictable.

Any idea how to sew the cut edges in such a way that it doesn't stop the knit from shrinking as it should? I don't have a serger. Double row of zig zag stitch?

If you swish a wool knit around in hot water and then put it in the dryer, it is going to shrink/felt no matter what.