So many agents focus more on the regular old home buyer.... and that doesn't always translate to understanding an investor mindset.But to answer your question you are likely looking at this backwards.

Make the initial purchase personally under my name, secure financing under my name, then quitclaim the property to the LLC?

Mortgages are the customary security instruments.

The act wasn't intentional, but should I still approach them about paying some of the bill or taking a portion of the bill from their security deposit?

Be sure you’re getting estoppel letters signed by each tenant indicating their current rent and security deposit along with leases.

I know most of the time they look at the security rather than the individual, but if you built rapport with loan officers or liquid individual (for lack of a better term) can you attain cash quickly this way?

You need this for your records to protect yourself in case the tenant claims they paid the old owner.Make sure the security deposit 100% transfers to you.

Income from flipping is subject to payroll taxes (social security and medicare).

However, she also said the child is having a hard time securing a down payment.

It's legal for regular business, but I have a feeling it is violating some type of antidiscrimination laws for HUD with respect to renting.