Starting a possibly infringing business, need help!

Okay so long story short I am thinking of setting up my own short term small business involving caps. We all know the 'New Era' caps with baseball, NBA, NRL etc team names/logos on the front. I currently have a source that sells basically unbranded versions of these caps for fairly cheap if you buy in bulk. What I would like to do is to start my own brand up, getting my source on board and embroid + sticker the caps with my own custom logo (whilst still keeping the team logos).

What I want to know before I go ahead and make a mistake is:

• Am I required to seek permission of any teams I use in order to use their logo/name?
• As the hats are obviously fake, once rebranded are these okay to sell?
• Due to cheap costs and this only being a small business I would be able to undercut New Era prices by over half, could this land me in trouble with them if it so happens that the business takes off?
• Finally seeing as I am based outside the US and would only sell within my own country is this likely to protect me from suing from US companies?