We can tackle each part separately. The .strftime('%b') method will extract the month abbreviation from datetimes. And we can use the .update method to set the range on fig['layout']['yaxis1'],...,fig['layout']['yaxis9'] which will allow you access each of the subplots, and we will want to set the parameter autorange to False. Putting this in a loop is best.

import numpy as np
import pandas as pd

from plotly.subplots import make_subplots
import plotly.graph_objects as go

df = pd.DataFrame({'Month': ['2020-01-01', '2020-02-01', '2020-03-01', '2020-04-01', '2020-05-01', '2020-06-01'],
          'Site A': [0.0006171, 0.0007480000000000001, 0.00041139999999999997, 0.0005422999999999999, 9.35e-05, 0.0011407],
          'Site B': [0.0003927000000000001, 0.0026, 0.0008041000000000001, 0.0005797, 0.0008789000000000001, 0.0004301000000000001],
          'Site C': [0.0075548, 0.0045815000000000005, 0.0033473, 0.0016455999999999999, 0.0023375, 0.00229],
          'Site D': [0.0007854000000000001, 0.0003927000000000001, 0.0013277, 0.0005235999999999999, 0.0008227999999999999, 0.0016082000000000002],
          'Site E': [0.0, 0.0007480000000000001, 0.0, 0.0015520999999999998, 0.0005984000000000001, 0.00014],
          'Site F': [0.0, 0.0007292999999999999, 0.0, 0.0002431, 0.0, 0.0],
          'Site G': [0.0006919000000000001, 0.0008976000000000001, 0.0005422999999999999, 0.0007667, 0.0008414999999999999, 0.0008],
          'Site H': [0.00257, 0.00324, 0.00512, 0.00197, 0.0009199999999999999, 0.0004301000000000001],
          'Site I': [0.0013277, 0.0, 0.0, 0.0, 0.0, 0.0013277]})

df['Month'] = pd.to_datetime(df['Month'])
df['Month'] = [mydate.strftime('%b') for mydate in df['Month']]
df = df.set_index('Month')

cols = ("Site H", "Site E", "Site B",
        "Site C", "Site G", "Site F",
        "Site D", "Site I", "Site A",
        )

colors = [['darkgreen','limegreen','lightgreen'],
          ['black','gray','silver'],
          ['darkred','tomato','lightsalmon'],
          ]

fig = make_subplots(rows=3,cols=3,
                    start_cell='top-left',
                    column_widths = [1200]*3,
                    x_title = 'Month',
                    subplot_titles=cols
                   )

fig.add_scatter(x=df.index, y=df['Site H'],  row=1, col=1, showlegend=False, line=dict(color=colors[0][0]),  mode='lines+markers', name='Site H')
fig.add_scatter(x=df.index, y=df['Site E'],  row=1, col=2, showlegend=False, line=dict(color=colors[0][1]))
fig.add_scatter(x=df.index, y=df['Site B'],  row=1, col=3, showlegend=False, line=dict(color=colors[0][2]))

fig.add_scatter(x=df.index, y=df['Site C'],  row=2, col=1, showlegend=False, line=dict(color=colors[1][0]))
fig.add_scatter(x=df.index, y=df['Site G'],  row=2, col=2, showlegend=False, line=dict(color=colors[1][1]))
fig.add_scatter(x=df.index, y=df['Site F'],  row=2, col=3, showlegend=False, line=dict(color=colors[1][2]))

fig.add_scatter(x=df.index, y=df['Site D'],  row=3, col=1, showlegend=False, line=dict(color=colors[2][0]))
fig.add_scatter(x=df.index, y=df['Site I'],  row=3, col=2, showlegend=False, line=dict(color=colors[2][1]))
fig.add_scatter(x=df.index, y=df['Site A'],  row=3, col=3, showlegend=False, line=dict(color=colors[2][2]))

fig.update_xaxes(tickangle=90, tickformat="%b", type='category')
fig.update_yaxes(tickformat=".1%")

tickvals = [i/1000 for i in range(1,9)]

# fig.update_yaxes(row=1, col=1, title='Success',    title_font_color="darkgreen", autorange = True, dtick=0.001)
fig.update_yaxes(row=1, col=1, title='Success',    title_font_color=colors[0][0], autorange = True, tickmode='array', tickvals=tickvals)
fig.update_yaxes(row=2, col=1, title='Status Quo', title_font_color=colors[1][0], autorange = True, tickmode='array', tickvals=tickvals)
fig.update_yaxes(row=3, col=1, title='Watch',      title_font_color=colors[2][0], autorange = True, tickmode='array', tickvals=tickvals)

# updates fig['layout']['yaxis1']... fig['layout']['yaxis9']
for i in range(1,10):
    fig['layout']['yaxis' + str(i)].update(range=[0, 0.008], autorange=False)

fig.update_layout(
    title='2020 Monthy Error Rate by Site',
    title_x=0.5,
    autosize=False,
    width=800,
    height=800,
    margin=dict(
        l=120,
        r=30,
        b=80,
        t=80,
        pad=0
    ),
    paper_bgcolor="LightSteelBlue",
)

fig.add_annotation(dict(font=dict(color="black",size=14),
                            x=-0.16,
                            y=0.5,
                            showarrow=False,
                            text='Error Rate (%)',
                            textangle=-90,
                            xref="paper",
                            yref="paper"
                           )
                  )

fig.show()