Box = usually a six-sided container made of cardboard (most often), wood, steel, plastic, paper, whatever.
Package = a container and its contents. Can be a box, envelope, etc.
Packet = a "package" usually an envelope with paper in it.
Many variations on the theme of these words.
Do you have specific questions on their usage?
I'd definitely call them "packets," too. I have heard some people use a rather odd-sounding (but logical) construction like this: "Could I have some more hot sauces, please?" (meaning "some more packets of hot sauce"). I haven't heard this with ketchup or mayonnaise, but I wouldn't be surprised if I did. "Ketchups" and "mayonnaises," though, are rather difficult to say, for some reason.
I have to agree with Sparky - sachet automatically makes me think of, for example, the dried aromantic flower known as lavender, and I don't even want to think about lavender and ketchup together. I accept that our brothers across the Pond have a different perspective, but it's really difficult for me to visualize, probably because of my hangup re. lavender and ketchup.
Wow. Another thing I've learned in the short time I've been here. Sachets of ketchup sounds disgusting. But then, I've heard BE speakers who nearly gagged when they first heard of biscuits and gravy. (NOT what it sounds like to Brits!)
I think this is why it's good to point out the differences.