I would start the whole process over with the tenant - application, screening, and approval.

But he kept pushing, and saying that they don't hold my application until I decide.

Anything more than a small application fee ($200-$300) and maybe an appraisal ($300-$500 if residential) and I would look elsewhere.

DSCR would be your best bet but again 25% down and a DSCR of 1.25% minimum.Do yourself a favor and do not let just any Lender/loan officer pull your credit just so they can take an application and have a possible deal.

What are the key differences in approval times and application processes between private money lending and traditional bank loans?

Is a PM allowed to share the tenants application including credit reports and other info that is collected during the application process?

Perform the same task in each app to ensure a fair comparison.Load a property with pictures and details.Market that property.See what your marketing looks like from the public's perspective.Submit a fake application to see how easy the process is.Run a credit/screening report on yourself.Enter a maintenance request, assign a vendor, and attach a fake invoice.Enter charges to the tenant's ledger.Enter recurring charges and automatic late fees.Sign documents electronically.Run owner reports.After testing a few apps, one should clearly stand out.

After everything was fixed and they listed the unit, apparently they claimed that I was not allowed to have any understanding of who would be renting my home due to potential bias and discrimination (this is the first time I have not been able to view applicants general profile (credit score/occupation(removing any identity markers)), but fine, I let them choose the tenant and so far there hasn't been any issues.

I wouldn't necessarily say you would need to increase the rent, unless the lease is already coming up on renewal and is due for an increase anyway.If you have a general application, have the incoming tenant just submit an application that way you can still ensure that they meet the criteria whether it be credit, background,etc... 

Subject to the requirements of the Bankruptcy Code, in the event the Tenant is adjudicated as bankrupt, (or makes an assignment for the benefit of creditors), this Lease, at the option of the Landlord, shall terminate upon 30 days written notice and the Dwelling Unit shall be surrendered to the Landlord, who reserves the right to repossess the Dwelling Unit subject to the applicable provisions of law.Should I follow the steps on the contract?