Due to size and breed restrictions I have, my risk is somewhere between non-existent to low.  

The BK is what's restricting your financing.  

@David Brizendine Los Angeles County...no city or lot ordinances or restriction.

@Brent Coombs There's a one year O/O restriction, or hold time where I carry it, after that it's up in the air...

A good real estate agent is also key, but remember they have a little bit of self-interest to get a property sold, in addition to some legal restrictions as far as what they are allowed to tell you, so while the majority will give tremendously valuable advice, I always suggest that you fly out to the property at least once before your diligence period expires to drive the neighborhood and get a feel for it yourself.

Custodians are allowed to restrict investments to something they control (such as gold, mutual funds, etc).

There is no restriction on making extra payments on a conventional loan.

Counties, cities, and HOA's all have their own laws, rules.....it will be the most restrictive one.

Can you please share more about the VA loan restrictions and buying in the Bronx?  

I became an investor to flex my creative muscles for getting deals done, thus I don't appreciate being restricted by rules and CC&R's that dictate how I run my business.