you can take care of the first 3 items with slicing and head:
df[df.A>=4].head(1)
df[(df.A>=4)&(df.B>=3)].head(1)
df[(df.A>=4)&((df.B>=3) * (df.C>=2))].head(1)
The condition in case nothing comes back you can handle with a try or an if...
try:
output = df[df.A>=6].head(1)
assert len(output) == 1
except:
output = df.sort_values('A',ascending=False).head(1)