Uiua

There's probably a good solution hiding in here, but this ain't it. I originally went for the combinatoric approach for part 3, but it was giving me an answer that turned out to be out by 8, so brute force came to the rescue.

Prep ← (
  ⊜□⊸≠@\n
  Pairs ← ≡⊂⊓(¤|⊜∘⊸≠@,)∩°□°⊟
  ⊃(≡(□Pairs⊜□⊸(¬⦷" > ")°□)↘1|⊜□⊸≠@,°□⊢)
  ↘¯1∧(⊂°□) ⊙["  "]
)
"Oronris,Urakris,Oroneth,Uraketh\nr > a,i,o\ni > p,w\nn > e,r\no > n,m\nk > f,r\na > k\nU > r\ne > t\nO > r\nt > h\n"
°□⊢▽⊸≡(/↧∊⊙(⧈₂∘°□))¤Prep # Part1 --> "Oroneth"
"Xanverax,Khargyth,Nexzeth,Helther,Braerex,Tirgryph,Kharverax\nr > v,e,a,g,y\na > e,v,x,r\ne > r,x,v,t\nh > a,e,v\ng > r,y\ny > p,t\ni > v,r\nK > h\nv > e\nB > r\nt > h\nN > e\np > h\nH > e\nl > t\nz > e\nX > a\nn > v\nx > z\nT > i\n"
/++1⊚≡(/↧∊⊙(⧈₂∘°□))¤Prep # Part2 --> 23
P ← Prep"Khara,Xaryt,Noxer,Kharax\nr > v,e,a,g,y\na > e,v,x,r,g\ne > r,x,v,t\nh > a,e,v\ng > r,y\ny > p,t\ni > v,r\nK > h\nv > e\nB > r\nt > h\nN > e\np > h\nH > e\nl > t\nz > e\nX > a\nn > v\nx > z\nT > i\n"
Keys  ← ⊙◌P
Names ← ◌ P
GoodNames ← (
  ˜▽Names≡(/↧∊⊙(⧈₂∘°□))¤P                 # Remove names that fail part2
  ▽⊸(¬≡(/+≡(⨬(0|≍∩⌞↙↧◡∩⧻∩°□)>◡∩(⧻°□))))⊸¤ # Exclude all names that are prefixes of others.
)
# Build up all names and count them. Yuk, slower, but righter.
⊙0GoodNames
≡◇([∘]
   ⍢(
    /◇⊂≡◇(□≡˜⊂⊙¤≡⊣▽⤚(=≡⊢)Keys⊸⊣) # Add next chars, collect.
    ⨬(∘|⊙+⟜⧻)⊸(≥7⧻⊢)             # Add to counts if long enough.
  | <11⊣△)
   ◌ # We don't need the data, just the counts.
)
/+