This is a classic "wide" to "long" reshape
problem (and your data--at least the sample data--are in a good form to do the conversion quite easily):
reshape(mydf, direction = "long", idvar="Date",
varying = 2:ncol(mydf), sep = "", timevar="C")
# Date C A B
# 1.4.04.1 1.4.04 1 5 10
# 2.4.04.1 2.4.04 1 2 11
# 3.4.04.1 3.4.04 1 4 14
# 1.4.04.2 1.4.04 2 1 13
# 2.4.04.2 2.4.04 2 2 12
# 3.4.04.2 3.4.04 2 4 15
# 1.4.04.3 1.4.04 3 2 4
# 2.4.04.3 2.4.04 3 0 5
# 3.4.04.3 3.4.04 3 3 8