We've rented to a family with no SSN, but required an extra month deposit and a co-signer.  

I was speaking with a co-worker and he had a similar plan.

A co-worker of mine did a 2-1/2  plus commute from Philadelphia to NYC everyday.

But it's critical to use someones borrowed credibility as a co-sponsor on my first deal.

I would do everything in my power to pay the back rent owed, stay in the $600 apt, put the kid in daycare, get a job, save enough for a co-op or condo downpayment and get an FHA loan. 

I've never personally done it, but a co-worker of mine bought a pressure washer and does pressure washing on the side...maybe $500-$1,000 investment (less if you don't go commercial grade) and you can get $100 per job on the weekend.

And are there other options like going month to month / a two month deposit / or a co-signer you would consider if you couldn’t get more color - or just pass all together?

We joined a co-op here in Orlando and after evaluating the cost as well as what value it would add to our home we decided to pass.

Alternatively, instead of lowering credit score criteria you can give them an opportunity to apply with a co-signer - but again, you have to apply whatever your rule is to everyone equally.

Could also look for a co-signer or go in on a loan with your roommate.