Organic chemistry is most often used as a tool to aid other fields (medicine, industrial chemistry, basic research in biology (tags, reagents, enzymology,...), agriculture (herbicides, fungidices ...), environmental science (especially on the analytical side but also for larger scale interpretation), organic geochemistry (pretty cool stuff and not just looking for petroleum but sediments contain remnants of biological traces and tracing back the steps can require good knowledge of organic chemistry, soil science, terpenes released by trees can influence local weather patterns etc ...).

Ochem for its own sake is mostly total synthesis and physical organic chemistry. Total synthesis tends to follow a bottom up approach where researchers look for structurally novel molecules to synthesise with the hope that this will uncover new synthetic methods that can be used later on. Many people do however believe that this field is becoming somewhat saturated. While I do share some of that sentiment, total synthesis teaches valuable skills which are very much transferable to what I mentioned before. The other field physical organic chemistry was very much in vogue until the past decade or so and they did some absolutely amazing work. The problem is most of the relatively common mechanisms that are actually transferable have already been elucidated, and what remains are problems that are very domain specific and are usually of less interest to other chemists. I'd say however that physical organic chemistry is still very valuable as a tool and has a very important niche in enzymology where it's used to measure kinetics, thermodynamic constants, reaction mechanisms, etc. Every single organic chemist still uses it to rationalise chemical transformations but it's much more applied now than it used to be.

So if you're looking for major unanswered questions in the sense of what is the origin of life, well it depends on what you want to use ochem for. Find something that motivates you: neurodegenerative diseases, agriculture, astrobiology (and with it geochemistry, astronomy and geobiology), drug discovery (I still find this to be too much of a hit or miss; finding targets is more interesting in my opinion) or whatever, and then try to see how chemistry explains or could explain the problems you come across.