At this point, I feel obligated to help him sell after that ordeal.

Property values had continued to drop in the area and in order to sell that home and move I would have had to pay someone to take it.

Then we either sell for profit or keep it as a rental.

Less expenses: Stop going out to eat so much, trade your car for a cheaper one, sell the $5000 golf clubs for $1000 ones, 1 vacation a year instead of 2 or 3 a year because "I deserve it", cut the grass yourself.I am being a little silly, but seriously get on a budget and see what you need to do to cut expenses or where you can work a little harder.

Assuming he/she does not sell the property at the end of 10 years, at that point will taxes be owed on the property's appraised value?

You'll probably end up cashflow negative and selling it to the next wishful out of state investor in 2 years after dealing with a bunch of BS and losing money.

Given current markets and my current rental cash flow on my one rental, what are some thoughts on selling the rental and using that as a downpayment for a nicer short term rental elsewhere?

I'd even recommend looking up on public records of some beat up small family in the area and calling them personally to see if they would sell you their property that needs work, especially if they owned it over 10 year, that usually means they are tired of it, and investing in bigger projects. 

However, we're considering selling to buy a primary residence where we live.

And maybe the listing broker knows that, that's why they think it should sell at a 7% cap rate, and maybe they are correct.