Hi Markus,
In general there will not be useful solutions for your scenario. With n=20 and 2 starts per round, you would need 19 rounds to get a balanced solution - at which point the competitors would have raced against each other 9 times. If you want a shorter schedule it would start to get messy, for example in 6 rounds you might be able to have everyone against each other at least once, but some pairs would have raced 5 times.
My feeling is that the seeds idea will makes things even harder, but I have nothing that could generate such a schedule to test my theory out.
Essentially the problem is that you want half the competitors in each race. If this fraction were smaller then there are options. Say 16 competitors with 4 starts of 4 per round.
Best,
Ian.