Can a character attack while insubstantial? If so, are there any limits on the type of attack?
Yes a character can attack while insubstantial(DDI). There are no limits on the type of attack.
The Rules Compendium p226 notes that creatures who are insubstantial take half damage from all attacks. (This also includes ongoing damage). There are no other restrictions or limitations placed on the Insubstantial creature.
Compare this to Gaseous Form(DDI). A monster in gaseous form is insubstantial AND cannot make attacks unless specifically noted.