+1 for the concern about sapwood in smaller diameter branches. Pegs are super-critical to the strength of the frame and they are tiny. Use the cleanest-grained, best-quality oak you can get your hands on. You'll only need a little. It's crucial that you use properly seasoned, stable oak too. If you can keep it inside for a month or so before using it, to ensure the moisture content is as low as it'll get, it would be worth doing.
As for making pegs, it is always going to be a time-consuming process. John's idea about hexagonal pegs is a good one - you can always go one further and shave it into 16 sides and it will be as good as round.
Another suggestion would be to make up a jig to hammer the pegs through. I've attached a
video of someone doing just that. It is essentially a tapered, sharpened tube made from decent grade of steel. I believe you need to roughly cut the pegs to fit before this though.