A lot of incoming mail is pushing up my system load too much, and there are many Exim processes. How can I control this?

Have you set any of the Exim configuration options that limit what it does under high load? For example, queue_only_load, deliver_queue_load_max? See the list in the section entitled Resource control in the manual. It sounds like a lot of simultaneous incoming mail pushes your system into uncontrolled overload. The multiple Exim processes are probably just multiple incoming messages. You can use the exiwhat utility to confirm this.