Jimmy.

Saving is done by going to the safe-house, please DO NOT EVER save while on-duty. It will corrupt saves. 
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For the GFWL update, please install these two: 
(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-)
(http://www.microsoft.com/en-us/download/details.aspx?id=5549)
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Where did you place the save-game files? Saves are located here: 
X:Users%username%AppDataLocalRockstar GamesGTA IVsavegames Also, when installing saves, I have found that the XLIVE Asi-loader is working more often than DSOUND asi-loader. 
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I thought you already had the mods, that's why I asked you to look in the readme. Sorry, my bad...
 
1-4: Those mods can be found here: ('?do=embed' frameborder='0' data-embedContent>>). 
1. It is indeed called ALPR
2. That is VDH Police Helper
3. That is the ELS, you need ELS enabled cars for it to work. Read the readme before launching the game. ELS enabled cars can be found in the topic I linked above!
4. That is also VDH Police Helper
 
As for the trainer, it's not "buggy" per say. You need to disable the trainer keys: 
Open the trainer with F3/F4 Go to Options Disable / Enable Trainer Keys. Set this to disable.  Now it should work! If you have any more problems, look in the forums or read my guide: ('?do=embed' frameborder='0' data-embedContent>>). 

Did you read the readme file? You need to press ctrl+z I believe. 

You are supposed to use the NumPad keys, make sure that NumLock is in 'on' mode on your keyboard. A lamp should shine somewhere on your keyboard, indicating it's active.