In the civil context, court costs are normally awarded to the prevailing party, meaning that the 'losing' party must cover them. Rule 54(d)(1) of the Federal Rules of Civil Procedure allows exceptions to this general rule via statute or court order.
If you have been making your payments regularly and on time, restitution and other court-ordered debt shouldn't show up on your credit report. However, once you begin missing payments, the court can sell your debt to a collection agency that will likely report the debt to the credit bureau.