 Here is a Python script that performs the tasks you've described using the `moviepy`, `youtube_dl`, `pydub`, and `Pillow` libraries. Please make sure to install these libraries using pip before running the script.

```python
import os
import subprocess
import moviepy.editor as mpy
from pydub import AudioSegment
from moviepy.video.io.bindings import mplfig_to_npimage
from PIL import Image

# Check if required files exist, download if not
video_files = ['video.mp4', 'audio.mp3']
for file in video_files:
    if not os.path.exists(file):
        try:
            subprocess.check_call(['youtube-dl', '-o', file, 'URL_TO_YOUR_VIDEO'])
        except Exception as e:
            print(f"Error downloading {file}: {e}")
            exit(1)

# Load audio, extract subclip, apply fade-in and fade-out, analyze period
audio = AudioSegment.from_mp3('audio.mp3')
audio_subclip = audio.subclip(0, audio.duration - 1)
audio_subclip = audio_subclip.fade_in(1000).fade_out(1000)
audio_period = audio_subclip.duration

# Load video, extract subclip, crop, analyze looping segment
video = mpy.VideoFileClip('video.mp4')
video_subclip = video.subclip(0, min(video.duration, 10))  # Adjust duration as needed
video_subclip = video_subclip.crop(x_start=100, y_start=100, x_end=video_subclip.w - 100, y_end=video_subclip.h - 100)
video_loop_start, video_loop_end = video_subclip.match_frame(video_subclip.frame, method='cross_correlation')

# Extract looping segment, slow it down, make it loop, create mirrored version
video_loop = video_subclip.subclip(video_loop_start, video_loop_end)
video_loop_slow = video_loop.set_duration(audio_period)
video_loop_slow_looped = video_loop_slow.set_loop(count=audio.num_frames // video_loop_slow.num_frames)
video_mirrored = video_loop_slow_looped.fl_transpose()

# Combine original and mirrored video segments, apply fade-in and fade-out, add audio
final_video = mpy.concatenate([video_loop_slow_looped, video_mirrored])
final_video = final_video.set_duration(audio_period * 2).fadein(1000).fadeout(1000)
final_video.audio = audio_subclip

# Create title and credits screens
title_image = Image.new('RGB', (final_video.w, final_video.h), color='white')
title_text = Image.fromarray(mplfig_to_npimage(title_font.figure(figsize=(final_video.w, final_video.h)))).transpose(1, 2, 0)
title_image.paste(title_text, (0, 0))
title_clip = mpy.VideoClip(title_image)

credits_image = Image.new('RGB', (final_video.w, final_video.h), color='black')
credits_text = Image.fromarray(mplfig_to_npimage(credits_font.figure(figsize=(final_video.w, final_video.h)))).transpose(1, 2, 0)
credits_image.paste(credits_text, (0, 0))
credits_clip = mpy.VideoClip(credits_image)

# Concatenate title, video, and credits into final video
final_video = mpy.concatenate([title_clip, final_video, credits_clip])

# Write final video to file
final_video.write_videofile('output.mp4', fps=30, audio_bitrate='128k', bitrate='1800')
```

Please note that you need to replace `URL_TO_YOUR_VIDEO` with the URL of the video you want to download, and you need to import `title_font` and `credits_font` which should be created using matplotlib.

Also, this script assumes that the video and audio files are in the same directory as the script, and it saves the final video as 'output.mp4'. You can adjust these paths as needed.