**Why does it crash?**<br>
Your code crashes blender because of a `EXCEPTION_ACCESS_VIOLATION`. For further information, see https://docs.blender.org/api/3.3/info_gotcha.html#help-my-script-crashes-blender. (you already know this, just for the sake of completeness)

Looking at the report it states:
```
Stack trace:
blender.exe         :0x00007FF616E36350  BLI_findstring
blender.exe         :0x00007FF611AE7CF0  IDP_AddToGroup
blender.exe         :0x00007FF612013EC0  RNA_property_string_set
blender.exe         :0x00007FF61658A070  file_browse_exec
blender.exe         :0x00007FF611DA05D0  wm_handler_fileselect_do
blender.exe         :0x00007FF611DA1F60  wm_handlers_do_intern
blender.exe         :0x00007FF611DA1390  wm_handlers_do
blender.exe         :0x00007FF611DA7920  wm_event_do_handlers
blender.exe         :0x00007FF611D8D820  WM_main
blender.exe         :0x00007FF610E82360  main
blender.exe         :0x00007FF617069750  __scrt_common_main_seh
KERNEL32.DLL        :0x00007FFB5D017600  BaseThreadInitThunk
ntdll.dll           :0x00007FFB5EF226D0  RtlUserThreadStart
```
Therefore I assume base on `BLI_findstring` the location of the `folder:StringProperty` has changed in memory after the selection, which then causes an `EXCEPTION_ACCESS_VIOLATION`.

Another attempt of explanation could be to add the following to the `folder:StringProperty`:
```
    def get_folder(self):
        return self.get("folder", "")
    def set_folder(self, value):
        self["folder"] = value
    folder: bpy.props.StringProperty(name="Choose Folder", subtype="DIR_PATH", get=get_folder, set=set_folder)
```
This will cause the following error:
```
Traceback (most recent call last):
  File "\Text", line 18, in set_folder
TypeError: unsubscriptable object
File "\Text", line 16, in set_folder
Traceback (most recent call last):
  File "\Text", line 16, in get_folder
```
because an instance of `SelectDirExample` is unsubscriptable to `self['key'] = value`.
This getter, setter implementation maybe prevent the blender from crashing, because the default and setter, getter function are overwritten.

**Solution**<br>
I don't think an easy solution like this is possible because you are using a dialog which is a floating window and the folder selection will overwrite the current floating window. Also take a look at the info area/info-log of Blender, the Operator isn't executed/registered. Which means it hasn't returned `{'FINISHED'}` and `execute` was never called.
Therefore, to implement something like this, you would need to call operator `SelectDirExample` again.

```python
import bpy
from bpy.types import Operator, PropertyGroup

class SelectDirExample(Operator):
    bl_idname = "object.select_dir_example"
    bl_label = "Select Dir Example"
    bl_options = {'REGISTER', 'UNDO'}

    instances = 0
    invoked = False
    
    def setter_getter(name: str):
        return {
            'get': lambda self: getattr(bpy.context.scene.selected_dir, name),
            'set': lambda self, value: setattr(bpy.context.scene.selected_dir, name, value)
        }
    
    input1: bpy.props.FloatProperty(name="Input 1", default=1.0, min=0.0, max=100.0, **setter_getter('input1'))
    input2: bpy.props.FloatProperty(name="Input 2", default=1.0, min=0.0, max=100.0, **setter_getter('input2'))

    def get_folder(self):
        return bpy.context.scene.selected_dir.folder
    def set_folder(self, value):
        props = bpy.context.scene.selected_dir
        props.folder = value
        if SelectDirExample.instances != 0 or not SelectDirExample.invoked:
            return
        SelectDirExample.instances += 1
        bpy.ops.object.select_dir_example("INVOKE_DEFAULT", input1=props.input1, input2=props.input2, folder=props.folder)
        SelectDirExample.instances -= 1
        SelectDirExample.invoked = False
        
    folder: bpy.props.StringProperty(name="Choose Folder", subtype="DIR_PATH", get=get_folder, set=set_folder)

    def invoke(self, context, event):
        SelectDirExample.invoked = True
        return_code = context.window_manager.invoke_props_dialog(self, width = 300)
        return return_code

    def execute(self, context):
        
        input1 = self.input1
        input2 = self.input2
        folder = self.folder
        
        # My processing goes here
        print(f"Folder location is: {folder}")
        # Finished processing.
        
        SelectDirExample.invoked = False
        return {'FINISHED'}

class SelectDirExample_Props(PropertyGroup):
    input1: bpy.props.FloatProperty(name="Input 1", default=1.0, min=0.0, max=100.0)
    input2: bpy.props.FloatProperty(name="Input 2", default=1.0, min=0.0, max=100.0)
    folder: bpy.props.StringProperty(name="Choose Folder", subtype="DIR_PATH")

def register():
    bpy.utils.register_class(SelectDirExample)
    bpy.utils.register_class(SelectDirExample_Props)
    bpy.types.Scene.selected_dir = bpy.props.PointerProperty(type=SelectDirExample_Props)

def unregister():
    bpy.utils.unregister_class(SelectDirExample)
    bpy.utils.unregister_class(SelectDirExample_Props)
    del bpy.types.Scene.selected_folder

if __name__ == "__main__":
    register()
```
To not cause an `EXCEPTION_ACCESS_VIOLATION` another variable is used to store the folder path (which could also be inside the `AddonPreferences`).
`setter_getter` is used to store the values into the same `PropertyGroup` as the folder. Otherwise, these values could not be moved along with the new operator call.
They would be their default values because somehow the `self` in the `set_folder` is not the same as the `self` in the `execute` or `invoke`.
A new instance with the stored variables will then be launched immediately, and it looks like before (only a movement in location of the dialog).
The `SelectDirExample.invoked` variable is need otherwise it would launch two dialogs if the folder parameter is used on calling the operator e.g., `bpy.ops.object.select_dir_example('INVOKE_DEFAULT', folder='<my_path>')`. This does not apply if the call is from `set_folder`.