Another thing the nurses handling the picc line should consider is changing out the port on the end. The luer lock ports themselves need to be changed every few days, or after using the port to give blood or to do a blood draw. Using a push-pause technique on the flush syringe in between solutions infused, also helps the port to clear itself of bits of precipitate. It's like, on 10cc flush, push 1-2 cc, pause, push 2cc, pause, etc. If the thing is clogged with, say, LR or D5 solutions, try pulling back the syringe and get blood in the line, then push it back in, repeat twice, then use a good saline flush. Blood itself will help clear some of the solution, but you have to do a good job of pushing the blood back out of the line.
Also, the patient's position and how their head is turned may influence the ability to draw blood back. And, if the patient's line flushes well but won't draw, an ultrasound wouldn't hurt just to rule out a nice little DVT forming around the line.
Every patient's lungs have different background things going on; it sounds like your son has some reason the lungs keep requiring mechanical help, it would be a good thing to ask either the pulmonologist or the nurse to explain it to you a bit more. Sometimes the staff doesn't explain much because they think a nurse knows what's going on. Tell them you really need the explanation.