How do I Iterate over vertices to perform operation on them and connected verts

Question

In my object I have a number of those kind of objects that represent road intersections.

1. I want a way to iterate through the vertices and find all intersections (any vertex that has 3 or more other vertices connected to it. All these vertices will go into "Intersections" vertex group.

import bpy

obj = bpy.context.active_object
mesh = obj.data
bpy.ops.object.mode_set(mode='OBJECT')

vg = obj.vertex_groups.new(name="Intersection_Points")

# Loop through all vertices in the mesh
for vert in mesh.vertices:
    connected_verts = set()
    
    # Loop through all edges connected to the vertex
    for edge in mesh.edges:
        if vert.index in edge.vertices:
            connected_verts.update(edge.vertices)

    if len(connected_verts) >= 4:
        vg.add([vert.index], 1.0, 'ADD')

2. Out of each Intersection, I want to separate each possible set of 3 vertices, in this example - I want it to end up with those in the picture below and then be able to perform an operation on each of those, or put each one into its own vertex group.

Answer 1

The first question
As Markus von Broady said, it's more convenient to keep the data so you can use it later. So I suggest to use a dictionary with an intersection vertex as key and a set of linked ones as value.

intersection_dic = {}

#do loop
    connected_verts.remove(vert.index)
    intersection_dic[vert.index] = connected_verts

import bpy

bpy.ops.object.mode_set(mode='OBJECT')

obj = bpy.context.active_object
mesh = obj.data

vg = obj.vertex_groups.new(name="Intersection_Points")


intersection_dic = {}

# Loop through all vertices in the mesh
for vert in mesh.vertices:
    connected_verts = set()
    
    # Loop through all edges connected to the vertex
    for edge in mesh.edges:
        if vert.index in edge.vertices:
            connected_verts.update(edge.vertices)

    if len(connected_verts) >= 4:
        vg.add([vert.index], 1.0, 'ADD')
        
        connected_verts.remove(vert.index)
        intersection_dic[vert.index] = connected_verts

The seconde question
You can use the itertools module to create sets of 3 vertex indices like your drawing. That module contains itertools.combinations.

import itertools

for vc, value in intersection_dic.items():
    print(f"[{vc}]")
    for va, vb in itertools.combinations(value, 2):
        print(f"{va} - {vc} - {vb}") 

outcome

[4]
8 - 4 - 1
8 - 4 - 2
1 - 4 - 2
[5]
0 - 5 - 1
0 - 5 - 8
1 - 5 - 8
[6]
0 - 6 - 8
0 - 6 - 3
8 - 6 - 3
[7]
8 - 7 - 2
8 - 7 - 3
2 - 7 - 3
[8]
4 - 8 - 5
4 - 8 - 6
4 - 8 - 7
5 - 8 - 6
5 - 8 - 7
6 - 8 - 7

About BMesh
You can easily get connected_verts using using bmesh.

intersection_dic = {}

for vert in bm.verts:
    if len(vert.link_edges) < 3:
        continue

    connected_verts = [e.other_vert(vert).index for e in vert.link_edges]

    intersection_dic[vert.index] = connected_verts
    dvert = vert[dvert_lay]
    dvert[group_index] = 1.0