Leverage the 0% down payment, but be mindful of funding fees.

If you don't have the "dough" you don't have the "go".Based on what you are saying you will need 20% down payment for a rental + closing costs + operating reserves (in cash).Most people will not have this--some will do what they need to do to get it--some will not, some can not.A good first step might be to save up as much as you can for a year in a special savings account--by living frugally for 12 months.Then explore down payment assistance programs in your area for first time home buyers of a primary residence--if you qualify (and the home is in a good area for your 'Family" vs Gunshot Alley, then buy a home for yourself and your family.Getting on the track of being a homeowner will give you some additional strength, after a few years of ownership, re-asses your situation.Or work 2 jobs and save the 20% down, closing costs and operating capital you will need, but if you do that and your wife divorces you for it, then you will be in a worse situation than you are now.In the mean time, [1] read Rich Dad Poor Dad, and [2] have fun with your family doing things together (while you have the time), rent a boat for an afternoon, go on a pic-nick, hike in the woods, etc...

I have no income as a student, but have money for a down payment.

The one caveat is that I will be unemployed during graduate school, and I will be solely surviving off of grad plus loans for tuition and mortgage payments.

But our lender is only offering us a renovation or construction loan, which I can understand.So my questions are: should we put down as much as possible to lower the monthly payment of the reno or construction loan? 

Live in them for 2 years or long enough to save up another 5% down payment and repeat. 

You will not get a conventional loan like owner occupied with lower down payment etc. you can get a loan but you will need 20-25% down. 

For a 1000 unit SFR portfolio in Tampa, you could have a portfolio manager in Cleveland, OH, a call in Center in Topeka, Kansas, remote customer service team members in Mexico or (offshore) in India, and a rental payment processing center in New York.

DSCR loans are based off of down payment, credit score and either actual or market rents so it helps to supercharge an investor's real estate goals and net worth.

The three pillars that qualify a DSCR loan, your credit score, loan to value, and the DSCR score ( gross rent / PITI(a)) - getting preapproved should be no problem as long as you have enough down payment, usually 20%, closing costs, and reserves. .