How do banks and fintech's issue credit cards on the back end? I'm looking for very granular steps.

There's no way to guess about this, as checks will vary across time. For example, you might get a Barclay Apple card + they might check Transunion today + Experian tomorrow or both. There's no way you can form any conclusion about this, which will last more than a few days.

I have implemented a number of different solutions for similar models and found that Stripe is best (http://www.stripe.com)

Emphasize the lack of transparency regarding the deposit reduction. If they initially agreed to a lower deposit and then reversed their decision, it's reasonable to dispute the charge. Highlight the absence of a clear refund policy and how it was communicated to you. If they assured you of a refund in case of non-qualification and you didn't provide confidential documents due to privacy concerns, you have grounds for a full refund. Document the communication regarding the application deletion before the stated deadline. If they prematurely terminated your application without giving you the agreed-upon time to consider, it supports your case for a refund. Express your dissatisfaction with their customer service and compare it to your experiences with other agencies. You have the right to choose which services to engage with, and poor service can justify seeking a refund. Privacy and security concerns are valid reasons to withhold confidential information. Emphasize that you were unwilling to compromise your personal information, and this should not disqualify you from a refund. Compile all relevant email communications as evidence for your credit card dispute. Clearly outline your points, focusing on their breach of agreement, lack of transparency, premature application deletion, poor customer service, and your legitimate privacy concerns.

Setup your business as a pass through. You run the transactions + charge a fee. Likely, for the businesses you mention, they'd be happy to pay a premium fee... as you will be the only payment processor to accept their transactions.

I don't think there's a best practice, similar to how much you should charge for your B2B SaaS company, cause it really depends on the market size, cost to deliver, and perceived monetary value you're offering via your API. That being said, the way I've baselined any pricing for my products has been to look at 100 companies that share similar customers and pricing model and either look on their website or email their sales team and ask directly. Even if you get 10% response that's enough to get you started.