There is nothing fishy or illegal about this the ability to rent it.Now what is fishy is that you said she is renting to her sister.  

My partner has much more financing ability then I do and we agreed that she would help finance our first property or first few to help our business get started.My question is; If I find a multifamily that I want to purchase and she lends me the money for the downpayment, will I still be able to get a loan for the property?

In essence you're exchanging the principal payments for the ability to reinvest that money into another purchase.Think about it this way - with the mortgage your ROI is 4.5% (cost of debt), BUT if you want to withdraw your own money you have to re-qualify and pay the bank to access your equity.

Having a family is a definite detriment to your ability to invest in most relationships.

Now, I find it hard to think that all of those people considering house hacking these properties had done their homework when making offers as we are seeing properties go above asking prices again.Now, if the numbers aren't working for me as a savvy investor whose play is to buy a four flat and keep one unit to airbnb/my friend and I to use when we come into the city, I have to wonder if all these house hackers are really considering what a swift economic downturn will do to their ability to keep the building.

The benefit to me:-Potential cashflow w/ minimal investment-Ability to potentially negotiate lower selling priceBenefit to Seller:-Ability to find appropriate 1031 property and not be rushed by 1031 timeline requirements-Peace of mind that mortgage will be paid and he can sell wheneverQuestions:How would I calculate the correct lease price?

You always want to have the ability to not take the rent so as to prevent the cure.  

You're being sold a bill of goods....

Bottom line as long as you have the ability to repay and the valuation of the asset is sound you won't have an issue finding a lender willing to work with you but your first question will be "do you offer non-recourse loans".  

Most people who are using HM are doing so because they either have maxed out their existing credit lines or don't have the ability to use bank money.