Can lymphocytes have proteins that are complementary to host cell antigens?

You only kill off the ones that are reactive to host antigens (read: the ones that would attack your own cells!). So normal lymphocytic function isn't affected.

Because of the nature of the immune system (your body never knows what's going to attack it, so it has to be prepared for anything), the antigen receptors on the cell surface are generated completely randomly - it's the scattershot approach. Fire as many different type of receptors at invaders as you can, and one of them is bound to match. Unfortunately, the side effect of that is sometimes you randomly create antigen receptors that are host cell reactive. These lymphocytes are "function checked" in the thymus, and if they're self-reactive, they get killed off. Sometimes they can slip through this process, though, which can lead to auto-immune disease!