Tax staff are very nice and friendly and answered all my questions. I will share the answers:
1- BPT tax will not apply to your out of state passive income (rent), However, If that number is at or above $50,000 then anything you do in New Hampshire will be taxed at 8%. Example, you make $50,000 on rent in Texas, you now bought a property in New Hampshire and make $10,000 on it a year, since you already hit the $50,000 mark (even if its from out of state) you will have to pay the 8% on the $10,000 in New Hampshire, (even though you didn't exceed the $50,000 in NH). the cost to you will be an $800 tax bill on the $10,000.
2-BET tax, If you have a company out of state this does not apply to you as long as all your income comes from outside of NH. However, if you open a branch in NH and your above the base tax threshold from your out of state business receits, you will now have to pay the BET tax in NH even if you didn't reach the threshold from inside NH. In reality, you will have to pay the tax since its based on receits even if you lose money on your NH branch.