I don’t bill lunch, but short breaks between stints of focused work are absolutely billed (especially if I’m in the client’s office, which I am much if the time, because I’m available at a moment’s notice.

I wouldn’t lower your rate… That doesn’t really affect the ethics of billing, or contractual obligations or anything. If the client is unhappy with what you’re doing it’s not going to matter whether you’re paid more or less, and vice-versa (if they’re happy with how you work either way doesn’t matter).