I am only free to say that I have been advised to turn the matter over to the proper authorities, which I have done and there is more than one agency currently looking at the matter.

Do these companies typically have 3rd party services/placement agency/some function in HR to provide relocation assistance to a new/relocating employee (within the US or international). 

You could waste money if your ads are bad, and PPC ads can get VERY expensive.You could always get a digital marketing agency in your area or anywhere to run PPC ads for you as well if you have the money to do so.

I have read that you can contact  the nearest hospitals  and offer services, so how is this done successfully out if state and what are other means for the example staffing agencies. 

If you have a contract with a listing agent, they are selling your house and you have no obligation to them if you go and buy a house unless you signed buyer agency agreement with them.Side note, anyone working for .5 percent isn't worth working with and you sound like the worst client for an agent in the world.

This family of three finds that they never hear back from the larger rental agencies even after paying for an application.

The NC Working with Real Estate Agents Disclosure describes agency relationships, and I’m sure I signed it without fully grasping how it might affect me.

- If I were to buy in Indiana, how should I go about finding a reliable property management agency?

If you think this is too much to handle by yourself, you'll still be better off spending your money on a digital marketing agency with a proven record, or a content strategist.

Got agency credit with a fixed rate for seven years, will probably sell before that point.