They said that was too low for them, so I offered $175,000 and they accepted. 

I looked at the rental trend over the past 10 years, I should be fine in year 7 for the refinance unless rates are totally out of this world.Your right those rents would be really low for Stamford. 

Instead of moving into another single-family home, consider house hacking a duplex or triplex, this allows you to use low down payment loans (FHA, conventional) and offset your mortgage with rental income, accelerating your path to financial freedom.

If you're willing to have roommates, your options open much more broadly, and you can potentially assume a loan with a low interest rate. 

But, this will be front-end weighted and gradually slow towards the back half of the year.

Option 3- If they have qualified for FHA; counter the offer to see if they can qualify for low downpayment conventional loan

Department of Housing and Urban Development (HUD) and Bankrate, the big perks of an FHA home loan are:Lower Down Payments: They typically require a smaller down payment than conventional loans, sometimes as low as 3.5% of the home’s purchase price.Lower Credit Score Requirements: They’re designed to help buyers with credit scores that might not qualify for conventional financing.

It’s just more than I can afford, and I don’t want to offend you with a low offer.”

Multiple people had some bad things to say about the area, but when I moved and realized that it was a low income area with a lot of homeowners, I was motivated to improve the property to match the effort my neighbors have put in. 

Would a low enough sales price make all of my concerns go away?