They only played those three you listed before Ichidai. Seidō was a seeded team for the Spring Tournament and played starting Round 3 (vs. Eigen). Then Round 4 vs. Houdai. Round 5 (which is quarterfinals already) vs. Kasuga. Round 6 (semifinals) vs. Ichidai.

So then Seidou got to bypass Round 1 and Round 2. And Round 7 will be the Finals (assuming they make it that far I guess)

I just thought Round 4 was considered the Quarter Finals. I know that's what it was in the Fall Tournament. Does it change depending on how many playing teams there are?

I think there are roughly the same number of teams competing in the Fall and Spring Tournaments. Not sure though why then are the number of rounds in the tournaments different.

The different is that in Fall Tourney, there are no seedings. This leads to school having to play block games before playing in the main draw. Thus, the main draw is shorter in comparison to the Spring Tourney which should more or less have the same number of participants but there are seedings depending on the performance of the team in Senbatsu and to my knowledge, there are no block games aside the official main draw; hence, there are one more round in the Spring Tourney main draw.