Typically what is in your resume is what you want the employer to know as you see it. So there is no harm in posting your resume. Online applications may not ask the relevant questions and may not present your application well.
So posting your resume has advantages.
Yes always have a few copies of your resume with you for the interview.
Bring also a list of references or letters of recommendation and only provide these when requested.
If it is an out of town interview, in addition to these hard copies bring along these documents in electronic form in your USB thumb drive. So you can print additional copies at their office or at the hotel should you need more of them.
When posting your resume on online make sure it is a reputable websites. Posting application and resume on certain websites can result in spam and junk mail.
In the US I would advice not to provide information like Social Security #, date of birth, age, bank account numbers, credit card numbers or anything online that will have the risk of identify theft.
I would not apply online if they were to ask these information.