It depends ;)
I don't tend to do a final sand for the very reason you mention. However there have been times when the finish hasn't been to my liking so I have sanded - but in these cases I use the finest grade paper and sand very lightly to just take off the drips and/or dust flecks that have settled on the wood.
So if the penultimate coat is good you might be OK without sanding, but sanding does provide something for the final coat to bind to if the previous coat was applied some time ago, so ultimately it will provide a more durable finish. If you are painting the second coat as soon as the first coat is dry then you don't need to bother.
Is it DIYable? Yes. No. Maybe?
You may find that you start the job, everything works out just right, and you complete the work in record time and under budget. More likely, you'll run into a bunch of unforeseen problems, take 5 times as long as you'd expect, and end up way over budget.
The best way to tackle DIY work that you are not sure about, is to have the resources available to quickly fix mistakes. In your situation, it might be a good idea to purchase a replacement window before you begin. If you damage the old window beyond repair, you'll simply have to install the new one. On the other hand, if the repair works out, you can return the new window you purchased.
It also never hurts to have somebody with knowledge in the type of work you're doing, so if you run into trouble they can offer guidance.
If you come across something you are really not comfortable with, there are always professionals available to do the work for you. You might also be surprised that they charge less than you might think (not always the case, but does happen).
Best Answer
The product I most often use for glazing wood sash windows (DAP 33 Glazing Compound) must be painted or it will shrink and pull away from the glass.
Unfortunately the stuff takes a while to firm up, like at least a week. If you mess with it before it skins over and firms up, it will deform. Also, because it is an oil-based product you have to paint it with oil based paint (or you can use oil based primer then paint over that with water-based paint). You should lap your paint over the line where the putty meets the glass, to seal the gap between glazing putty and glass.