A Haskell solution, though I'm sure someone can do better ;)

Perhaps I'll clean it up using printf/format[0] later.

    {-# LANGUAGE OverloadedStrings #-}
    import           Control.Monad     (liftM)
    import qualified Data.HashMap.Lazy as H
    import           Data.List         (sort)
    import           Data.Maybe        (fromMaybe)
    import           Data.Monoid       ((<>))
    import qualified Data.Text         as T
    import qualified Data.Text.IO      as TIO
    import           Prelude
    
    main :: IO ()
    main = do
      gradesMap <- liftM
                   (H.fromList . map ((\(x:y:_) -> (x,y)) . T.splitOn "\t") . T.lines)
                   (TIO.readFile "example.txt")
      let failingNames = H.keys $ H.filter ((== F) . T.take 1) gradesMap
          gradeTotals = foldl
                        (\m k -> H.insertWith (+) (T.take 1 k) 1 m)
                        H.empty
                        (H.elems gradesMap)
      TIO.putStrLn $ "Zsófia's grades: " <>
        fromMaybe "N/A" (H.lookup "Zsófia" gradesMap)
      TIO.putStrLn "List of students with a failing grade:"
      TIO.putStrLn $ "  " <> T.intercalate ", " failingNames
      TIO.putStrLn "Distribution of grades by letter:"
      mapM_ (\(k,v) -> TIO.putStrLn $ "  " <> k <> ": " <> (T.pack . show $ v) <> " students")
        (sort . H.toList $ gradeTotals)

0: http://hackage.haskell.org/package/formatting

A working version with some stuff improved - groupby implemented, len() instead of sum(), a less squished version of printing, "student(s)" added:

    #!/usr/bin/env python3

    from collections import defaultdict

    def invert_dict(data):
        res = defaultdict(list)
        for k, v in data.items():
            res[v[0]].append(k)
        return res

    with open('data.txt', 'r') as f:
        data = dict(x.split() for x in f if x.strip())

    print("Zsófia's grade:", data["Zsófia"])
    print("List of students with a failing grade:")
    print("  " ,end="")
    print(*[x for x in data.keys() if data[x] == 'F'], sep=", ")
    print("Distribution of grades by letter:")
    grouped = invert_dict(data)
    for k, g in sorted(grouped.items()):
        print("  {}: {} student{}".format(k, len(g), 's' if len(g) > 1 else ''))

No need for `invert_dict`; you're just counting so `collections.Counter` would work fine:

    from collections import Counter

    with open('data.txt', 'r') as datafile:
        students = dict(line.rsplit(maxsplit=1) for line in datafile)

    failing = [name for name, grade in students.items() if grade >= 'E']
    grade_counts = Counter(grade[0] for grade in students.values())

    print("Zsófia's grade:", students["Zsófia"])

    print("List of students with a failing grade:")
    print("  " + ", ".join(failing))

    print("Distribution of grades by letter:")
    for grade, count in sorted(grade_counts.items()):
        print("  {}: {} student{}".format(grade, count, 's' * (count != 1)))