10oClock

Try removing your car mods.

Can you please check if it is removed? "Think" won't cut it. 

And is xlive.dll and all its dependencies removed? 

Post another screenshot of your directory. Also, make sure no xlive remains. 

Install this (http://www.microsoft.com/en-us/download/details.aspx?id=5549) and this (http://support.rockstargames.com/entries/484496-Grand-Theft-Auto-IV-Patch-7-Title-Update-v-1-0-7-0-English-1-0-6-1-Russian-1-0-5-2-Japanese-).
You do not have dsound.dll in your game directory btw. 

Alright "dude", please post a screenshot of your GTA IV directory. 

How is that relevant to you deleting xlive? 
I'm on whenever I feel like it :) 

No, you remove xlive completely. 
No, I do not help via teamviewer. This is a very simple task :)

XLIVE.dll will disable GFWL.
1. Make sure you have the latest patch.
2. Make sure xliveless is removed completely.

Does your home key work outside of the game? 

Then I do not know what is wrong. You're probably doing something wrong. :/

Mind telling me what you have tried? Just saying that you have tried a lot of things doesn't really help...
Anyway:
1. Install the latest GTA IV patch from R*: http://support.rockstargames.com/hc/en-us/articles/200145406-Grand-Theft-Auto-IV-Patch-7-Title-Update-v-1-0-7-0-English-1-0-6-1-Russian-1-0-5-2-Japanese-
 
2. Then install the following redists: 
http://www.microsoft.com/en-us/download/details.aspx?id=30679
http://www.microsoft.com/en-us/download/details.aspx?id=30653
http://www.microsoft.com/en-us/download/details.aspx?id=35
http://www.microsoft.com/en-us/download/details.aspx?id=5549
 
Lastly, run the game as administrator. Go to the GTA IV folder, right click on GTAIV.exe, Properties, Compatibility and select 'Run this program as administrator', do the same exact thing to the LaunchGTAIV.exe.

Gonna play some LCPDFR with 10oClock. Please add me GFWL: Trep Leedle
 
We're gonna use teamspeak

Good luck man! :)