First, I guess if you have not done so, send an apology letter to your X employer stating the issues you may have caused them because the intent is of selfish-act to land that job (sorry for the term but honestly this is what I can see you have done). They may respond or not it does not matter now at least you have the courage to inform your mistakes. Take the ownership of that. For the next employer or your prospective company, definitely submit all originals. It may not guarantee your BGC (as my understanding this means background check?) but if asked what was the reason you are looking for a new job, you may tell honestly. Most employers like honest candidates. It's a risk but as I said you must take the ownership of your responsibility. For the feedback by your X employer, you may never know once you send the apology letter. It may turn things around for you. But then again you should not expect a good feedback from X employer once they do background checks, let's hope for the best though.
I wish you the best of luck!
From Bangladesh, Dhaka
The responses of various companies are also different.
If the matter comes to the hr department and the person processing the query remembers that there was a previous fraud complaint, he may tell that. I which case the company will take a call based on their internal procedure. If he does not disclose but only confirms that he was relieved in specific date, then you are safe
From India, Mumbai