Yes, this is exactly what I'm asking: What is the best tool to use to trace over this? I have no desire to draw up the schematic as it's irrelevant to manufacture.
Unless you have zero interest in ever designing a PCB now or in the future, then you absolutely should draw the schematic first. Schematic->PCB layout->Gerber->Manufacturing is THE process for making PCBs, so a) it's the process the software is designed around, and b) doing the schematic means you CAN'T wire the PCB wrong. I cannot underscore enough how important (b) is! So if you have even the slightest interest in electronics, learn the process, you'll be doing yourself a favor.
Not necessarily. There is no right or wrong way in engineering, only more or less appropriate for a particular situation.
Doing the schematic first can impose unnecessary constraints as choices are nailed down too early in the process. In many cases there is a lot of scope to swap pins & devices ( MCU I/Os etc.) and the optimal arrangement can be far from obvious until you start looking at the physical layout.
I never start from a schematic, instead I create nets as I go along during placement, and to a lesser extent, routing - put some or all of the parts down, create nets for all the things that are fixed, then see what the best options are.
There is still a netlist, it's just created during the layout process. (I use PCAD2006, which allows nets to be created during layout - I don't know how common this is in other PCB software).
For a job like this, where you already have a layout, starting with a schematic would be a waste of time unless you intend to make substantial changes.
As mentioned above it's not worth the effort to convert from a scan, and much quicker to re-draw it from scratch in a PCB package.