I wouldn't use "under-confident." I'd either do as Su.Ki says--use a plain, clear phrase: "he lacks confidence," "he isn't confident," or look for a completely different word that has the opposite meaning. The ones that come to mind: "timid," "uncertain," "unsure," "insecure," "vacillating." Let's see what an online thesaurus suggests: http://www.thesaurus.com/browse/confident Of the ones it lists, I like "doubtful" and "unsure."

You can, but it's not very common. You can also say 'He lacks confidence'.