Only your doctor who diagnosed you will be able to answer that. No one here knows your health staus. Compression socks are not compression stockings. They can be above the knee or as high as the groin they are put on while you are still in bed with your leg raised and someone else should put them on for you. Never put them on after you've walked around the house or whatever you do in the morning. Before you do anything they need to be put on. They can help prevent blood clots. A blood clot can travel to your brain or your heart and KILL IN A SECOND OR TWO. IS IT WORTH IT? NO. CALL YOUR PHYSICIAN.
Yes, there is no reason not to fly after a lung biopsy.
Absolutely agree with gymdandee. This is definitely a question for your physician.
Speak to your doctor first! As for the compression stockings ask your doctor what compression size he/she recommends and even when on a plane you should always get up and walk every 1-2 hours.