Before I touch on your questions, it's worth noting that rather than simply drawing chests from singular references, what will be much more valuable is to create your own chest designs, using things like this as reference and combining different elements together. Of course if you've primarily been copying references thus far, that's not a big problem, but you will get a lot more out of the exercise if you focus on the design aspect of it, and creating your own, as that is generally what my feedback focuses on.

As to the lid of the chest in your image, it's less a question of how you should tackle it, but more a question of what each approach would leave you with. Constructing each of these forms using explicit markmaking (drawing the structures directly), will allow you to describe those forms more completely, but will also result in a greater concentration of visual information in that area, which will draw the eye. Employing implicit markmaking (drawing the shadows those forms cast) will allow you to better control how dense your details end up being, and therefore how much that area draws the viewer's attention, but will also make it less likely that the specific nature of those forms will come through.

Either one is an acceptable choice, it comes down to what your priorities are, and what information you want to convey - which is why looking at it from a design standpoint, of creating something new rather than simply conveying what is in front of you - is a much more useful way to approach this problem.

And as to your second question, that depends on the limitations under which you're working. The general restrictions of our course (which apply more loosely to this work, but it's still the context in which the challenge is framed) doesn't give us any tools for capturing surface colour or patterns. The focus is entirely on form and structure, and this kind of polished wood is, by intent, not going to be forthcoming in making the fact that it is wood clear without being able to rely on those kinds of elements. That's not to say there's nothing you could convey - the kinds of bevel work used throughout the structure is pretty commonly seen in wood, and adding little defects on some of your protruding elements (where the chest might have hit other things in the course of its existence) could suggest the softness of that material, but ultimately not everything can be conveyed as strongly when you're working under set limitations.

That doesn't mean your result would be lesser for it - just that it wouldn't make sense to worry too much about that aspect of what the result conveys. In a normal context where we're making our own decisions from top to bottom on how to tackle something, we'd first consider what our priorities are (in terms of what information we wish to convey) and then choose the tools/medium/restrictions that suit them best.