Once you reach the point where you have been delinquent for 180 days, by law, your credit card issuer will charge off your account, which will ruin your credit score and will remain on your credit report for up to seven years.  After your account has been charged off, your credit card issuer will likely sell the debt associated with the account you had with them to a third party, external collections agency.   Keep in mind that once you are near, or have reached the point where your account is charged off, you may be threatened with a lawsuit or actually sued if there is a reasonable belief that you have the money to pay off a significant portion of your credit card debt.
The bottom line is that not paying your credit card debt will lead to your credit being completely destroyed.  Given the adverse effects, if you have the money to pay your credit card bill you should do so and do so on time.  If you don’t have the money to handle your credit card debt, call your credit card company and try and work out an arrangement that is both affordable for you and reasonable to them.

Yes.  They can do both.  If your credit card issuer suspects that you have the money to repay your account they can and will sue you.  This usually happens you’re your credit card balance is $1000 or above. In most cases, if you become severely delinquent on payments, your credit card company will close your account and put it into collections.  But the worst thing that can happen if you don’t pay your credit card bill for a long period of time is your credit score being destroyed.  This will make it very hard for you to get approved for credit cards or any other type of loan in the future.