For several reasons (property taxes, politics), I would prefer to begin real estate investing somewhere other than locally. 

I know a common practice (at least for syndications), is to create a separate LLC for the property and then raise capital once the deal is already under contract.

If not I wouldn't bother selling as there is an incredible amount of money lost in buying and selling properties, so you have to make sure you have a plan to maximize the equity and actually have enough. 

Yes this is possible.Go check with your original lender on the first property if you had one. 

I've purchased all of my properties while out of state or out of country.

If you decide to sell and find an investment property with a greater cash flow potential.A 1031 exchange would allow you to indefinitely all of the tax and use it to your advantage to reinvest into larger nicer property/properties.

We worked with a lawyer over there who is a US ex-pat who created and registered a local LLC for us, which owns our properties.

New to Arizona and I am seeing a few properties with next gen configuration.

Is this property in TN?

How should I phrase my interest in the property to avoid coming across as pushy or insensitive?