Row merging based on two conditions

I would start like this, this is assuming you only want one row/record per farm-shed.

import pandas as pd
import functools

# dataframe is df

# First combine horizontally: get the average weight into one column

weight_cols = [col for col in df.columns if col.startswith("Average weight")]
df2 = (
 df.assign(**{
    'Average weight': lambda df: functools.reduce(pd.Series.combine_first, [df[col] for col in weight_cols])
 })
 .drop(columns=weight_cols)
)


# Get first value from each column in each group

grouped_by = ['Farm', 'Shed']
remaining_cols = df2.columns.difference(grouped_by, sort=False)

(df2.groupby(grouped_by)
    .agg({col: "first" for col in remaining_cols})
    .reset_index())

If you want to safeguard against duplicated data, more checks are needed.