Consider the following:

import matplotlib as mpl
import seaborn as sns
import numpy as np

A = np.random.randint(0,100, size=(20,20))
mask_array = np.zeros((20, 20), dtype=bool)
mask_array[:, :5] = True
cmap = mpl.colormaps["viridis"]

newcolors = cmap(np.linspace(0, 1, 100))
newcolors[:10] = np.array([1,1,1,1])
newcolors[90:] = np.array([0,0,0,1])
newcolors[20] = mpl.colors.to_rgb('tab:orange') + (1,)

newcmap = mpl.colors.ListedColormap(newcolors)

g = sns.heatmap(A, cmap=newcmap, mask=mask_array)
# Set color of masked region
g.set_facecolor('lightgrey')

Sample result:

The following (appended to the above script) will make it so that the colorbar has a visible outline.

cbar_ax = g.figure.axes[-1]

for spine in cbar_ax.spines.values():
    spine.set(visible=True)

Sample result with outline:

In order to mask the colors of the heatmap, but not show an updated colorbar, set cbar=False, and then attach a custom colorbar, as shown in Standalone colorbar.

g = sns.heatmap(A, cmap=newcmap, mask=mask_array, cbar=False)

# add a new axes of the desired shape
cb = g.figure.add_axes([0.93, 0.11, 0.025, 0.77])

# attach a new colorbar to the axes
mpl.colorbar.ColorbarBase(cb, cmap='viridis', norm=mpl.colors.Normalize(10, 90),  # vmax and vmin
                          label=None, ticks=range(10, 91, 10))

The default cmap is sns.cm.rocket. To reverse it set cmap to sns.cm.rocket_r

Using your code:

cmap = sns.cm.rocket_r

ax = sns.heatmap(cm_prob,
                 annot=False,
                 fmt=".3f",
                 xticklabels=print_categories,
                 yticklabels=print_categories,
                 vmin=-0.05,
                 cmap = cmap)

If you use a BoundaryNorm, you can give colors for the ranges between the boundaries. To get 4 ranges, you need 5 boundaries. One approach is to add one extra boundary at the end. In the question it is unclear what you want to do with colorvalues that don't coincide with a boundary. In the code below, the color is used for a boundary value and the range up till the next boundary.

import seaborn as sns
import numpy as np
import matplotlib
import matplotlib.pyplot as plt

matrix = np.array([[149030, 34], [7442, 12]])
boundaries = [value for value in matrix.flatten().tolist()]
list.sort(boundaries)
colors = ["#90AFC5", "#336B87", "#2a3132", "#763626"]
norm = matplotlib.colors.BoundaryNorm(boundaries=boundaries + [boundaries[-1]], ncolors=256)

cmap = matplotlib.colors.LinearSegmentedColormap.from_list("", colors)

fig = plt.figure(figsize=(6, 6))
ax = plt.subplot()

annot = np.array([[f"{matrix[0, 0]}", f"{matrix[0, 1]}"],
                  [f"{matrix[1, 0]}", f"{matrix[1, 1]}"]], dtype=object)

sns.heatmap(matrix,
            annot=annot,
            annot_kws={"size": 11},
            fmt="",
            ax=ax,
            cmap=cmap,
            norm=norm,
            cbar=True,
            cbar_kws={'format': '%.0f%%', 'ticks': boundaries, 'drawedges': True},
            xticklabels=False,
            yticklabels=False)
plt.show()

If you set vmin and vmax the mapping from numbers to colors will always be the same, also when some values are missing from the data.

Some ideas:

• Use a ListedColormap with the given list of 6 colors
• Set vmin=-0.5, vmax=5.5 to have the values 0,1,2,3,4,5 nicely at the center of each color.
• As seaborn by default makes the colorbar border invisible, and one of the colors is white, the colorbar border could be turned visible.

import matplotlib.pyplot as plt
from matplotlib.colors import ListedColormap
import seaborn as sns
import pandas as pd
import numpy as np

cmap_dict = {0: '#FFFFFF', 1: '#ff2a00', 2: '#ff5500', 3: '#ff8000', 4: '#ffaa00', 5: '#ffd500'}
cmap = ListedColormap([cmap_dict[i] for i in range(6)])
df = pd.DataFrame(np.random.randint(0, 6, size=(15, 15)), index=[*'abcdefghijklmno'])

ax = sns.heatmap(data=df, cmap=cmap, vmin=-0.5, vmax=5.5, yticklabels=True, xticklabels=False)

for spine in ax.collections[0].colorbar.ax.spines.values():
    spine.set_visible(True) # show the border of the colorbar
plt.tight_layout()
plt.show()