If need matched columns only for 1 values:
df = (df.set_index('name')
.eq(1)
.dot(df.columns[1:].astype(str) + ',')
.str.rstrip(',')
.str.split(',', expand=True)
.add_prefix('c')
.reset_index())
print (df)
Explanation:
Idea is create boolean mask with True for values which are replaced by columns names - so compare by DataFrame.eq by 1 and used matrix multiplication by DataFrame.dot by all columns without first with added separator. Then remove last traling separator by Series.str.rstrip and use Series.str.split for new column, changed columns names by DataFrame.add_prefix.
Another solution:
df1 = df.set_index('name').eq(1).apply(lambda x: x.index[x].tolist(), 1)
df = pd.DataFrame(df1.values.tolist(), index=df1.index).add_prefix('c').reset_index()