You should make it policy to NOT allow anyone to move in without paying first month's rent in it's entirety PLUS the security deposit. 

@Johnery LaurimoreRecommend exploring as many sources as possible to get referrals AND cross-reference them to get as much accurate information as possible.Check out NARPM.com, BP’s Property Manager Finder, etc.Also, encourage you to learn from the mistakes of others - by reading posts here on BiggerPockets about owners not having their expectations met by their current Property Management Company.

To provide some context, she initially approached our company to manage her HOA.

Account Closed is correct, absolutely do both.. especially if youre just starting out even Grant Cardone started with a SFR. but for the sake of his question lets say you have 100k saved up for a real estate endeavor TOTAL, and you find a SFR that fits the 1% ratio (100k house that brings in 1k rent) that is doable if you calculate it out that would equal a 8.2% cash on cash IF it stays at a 90% occupancy rate. on the other hand if you invest that 100k into a limited partnership with a company that invests in value add apartments will now your cash on cash can be a preferred 10% with a target of 16-20% IRR which would essentially double your money in 2-5 years.. in this scenario the SFR would take sweat equity from you and risk while only returning a measly 8% CoC while the MF would be completely passive allowing you to learn and grow without hindrance with a 10% CoCnow we are over simplifying but I hope this made sense.. cuz my brain hurts ;D

We recently had a unique offer from an elderly care company to rent our home to operate their business out of.

HAve you had any good experience with any company?

I have a separate company, Tax ID and account set up for my short term rental company.

c) Can income (rents paid to me) and expenses (e.g. electrician) be paid directly to the holding company? 

Pete,Apartments are a class of business that some Insurance companies like and others avoid. 

When companies or properties are not self sustaining, you would initiate an "owner contribution" to cover any expense the business/rental property could not cover.