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Fifty four words put G first and O third, and twenty of them repeat a letter. The GOO branch is responsible for most of that.
R takes the second box in eighteen words, L in sixteen and O in fourteen. Together those three cover forty eight of the fifty four, and almost nothing else appears there at all.
GROUP, GROAN, GROVE and GROWN come out of the R branch. GLOVE, GLORY, GLOOM and GLOAT come out of the L branch. The two behave differently at the end, with the GRO words leaning on N and the GLO words spreading wider.
GHOST and GHOUL use an H that does almost nothing, GNOME opens with a silent G, and GEODE and GAOLS put a vowel in the second box. Each of those is worth more than another guess inside the GRO branch.
S ends sixteen of these, thirty percent, and Y another nine. The Y is higher than usual and comes mostly from the GOO words, where GOOEY, GOOFY and GOODY all land on it.
Because the second and third boxes are both O by definition of that branch. Every one of those fourteen words carries the repeat automatically.
GROUP is the more common word. GROSS doubles the S, which puts it in the repeat group and makes it easier to rule out when a double is impossible.
Yes. It runs G, H, O, S, T, so the O sits third. GHOUL is its only close relative here.