If its a shoe... you aren't barefoot. Even if its V.F.F.'s.
I train barefoot, but only my striking and grappling since you don't compete with shoes on.
I wouldn't recommend lifting barefoot ever, just doesn't sound smart.
Running barefoot, unless its a clean beach, also wouldn't recommend. If for no other reason than to avoid stepping on something sharp you failed to see and slicing your foot.
Isshinryu Karate 2nd degree Black belt.
Neko-Ryu Goshin Jitsu 2nd degree Black belt.
Jujitsu 2nd degree Black Belt.
Isshin-Jujitsu 2nd degree Black belt.