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are you on probation or parole - because that is going to restrict your movements. additionally, getting a passport is going to be tough as a sexual offender - you can get the passport, if not on probation or parole - but the US will notify those countries of your impending ventures - those countries can preclude your entrance
no. there is really no way to tell - you can check the embassy website and see what it says about admittance. and, yes - you will need to disclose your record on the passport application. but if no parole or probation, then you should be granted one. as to leaving the state - as long as you do not take a permanent residence - then you should not need to register in those states. but you will want to check the website of the sheriff's office in each county you plan on visiting to make sure they do not required notice of your arrival - they might. if you violate thier specific laws, then you can be charged - each state is different