Help: job application for an on site interview

I just started a new job and got an on site interview after the phone interview (which happened before starting the new job). The HR sent me application form today for the on site interview. Should I put the new company in the form? If I do, no hope for the offer. If I don't, will they find out in the background check anyway? Does anyone know what exactly the company checks in the background check? The company for the interview is a international large company. The background check is supposed to be very strict.

Recently, there were some recruiters contacted me for new opportunities as my resume is still on the internet. Should I tell them that I already started a new job? One opportunity is really good. But if I let the recruiter know my current status, there is no hope for the opportunity either.

Thanks a lot for your suggestions!