I wanted to be named as additional or co-insured, but was told by insurer that don't do that because I was already covered as a member of the LLC (sole member). 

@Tia Mitchell right but you are looking for additional funds that the lender won’t cover correct... 10%-25% or did I misread what you are looking for.

We thought the properties were both vacant, so we could just close, rehab, then either rent & refi and get all our money out, as even a 70% LTV would be enough to cover our purchase + rehab.

Great post @Thurman Kennedy, my first real estate deal was a duplex house hack here in San Antonio.Since I am a Realtor & I used an FHA, 3% loan, I was able to buy it with only $500 out of pocket because the seller covered some closing costs which were wrapped into the loan and I received a 3% real estate commission at closing.In hindsight, I wish I had purchased a 4-unit instead of a duplex.

Where they long term rent it from you and cover utilities and such? 

There are a couple of ways you can hedge this, the first one is via an umbrella policy by your insurance company (you can get one that covers all of the assets), two the fact there is a mortgage means that you do not own the asset free and clear and have only limited equity so if some tries to sue you it may not be worth it.   

I think that covers the basics.Hope this helps and hope you find a great deal soon!

A very small price to pay to make sure you are covered in all sorts of liability issues.

Seems to me that any basic lease should cover renewals, notices, etc...

I told them that we will take out of the deposit to cover any vacancy and she's been cooperative, trying to help find another tenant, etc so we're trying to keep it from becoming contentious.