"Dykking" is a pastime, or even a job. "å dykke" can refer to that, or the simple act of diving.
This sentence could be referring to either something she habitually does (for a living or in her spare time), or something she is doing right now, and thus translates to either the simple present or the present continuous in English depending on the context.
Beware that "å dykke" in Norwegian refers to the diving one does underwater. The act of plunging headfirst into the water is "å stupe".
This one is tough. To a native English speaker, things are generally only expensive if they are for sale, or you are discussing a sale price. If selling or replacing (by buying a replacement) is not the context, we almost never use "expensive". We would not say she was diving for expensive things, unless they were things that she or someone she knew had lost them overboard and she was trying to recover them. :) Or if she was going to jack up the price for a prospective buyer. Without an immediate sale or replacement purchase, we just wouldn't use "expensive."
I think it's a matter of a shift in meaning in English due to French courtly influence: "dear" probably became used as a direct substitute for French "cher", which DOES have either meaning - sentimental or monetary - pretty equivocally. But "dear" comes from the same background as "dyr", meaning, literally, an animal - wealth in old Scandinavia was expressed in livestock.