Are there any CC&R's on the property that will restrict the style, type, size of building or the end business?

I convinced them to advertise "well-trained family dog considered, breed restrictions, no puppies", as I always do on my own, and it seems to be working.  

I am currently under contract with a condo in North Palm Beach Florida that restricts the ability to lease within the first year of owning the property.

By doing it that way, the Fannie Mae restriction of re-selling the property for no more than 120% does not apply to you.

If I have to wait one or two years before buying each property then my real estate plan is going to hit a brick wall.Is it possible that banks add additional restrictions on top of Fannie Mae?

I don't know that they can prevent the owner from renting it (maybe they have the right to disapprove all tenants in that case) but they Do have the right to restrict the occupant from common amenities (pool, gym,etc). 

Are there any restrictions/ obstacles to haveing properties in the same LLC in 2 different states?

FHA doesn't have any restrictions on resale anymore, but there are a few things to watch out for:-  Even though FHA doesn't have the restrictions, some lenders/banks still impose the restrictions. 

An offer that a Seller wouldn't reject for a lower cash price that did not include any contingencies and/or Earnest Money restrictions as noted above?

The c offering lets you advertise nationwide, and there is no restriction on having unaccredited investors view your ads, including in emails and on social media.