Does the bride and groom exchange vows when getting married by court? Do they give one a other the rings?

There must be some exchange of vows to each other to be married. People marry each other, the court merely validates it. The court doesn't say I marry you, it says, I pronounce you man and wife or couple or what ever is said in the state where you are married. Rings are not necessary.

The vows are "I do". The judge or JP may allow you to exchange vows you have written or they may not. Usually rings are exchanged, but are not required.

It all depends on the presiding judge.

Each judge has its own, some even do it in small groups. In others all the people to be married are in the waiting room and go into the judges office. It is almost always done in the office, not court room, Most are informal, some wear suits, most I see causual or even jeans.

Each judge has his own small vows, most he says your name, asks to see your ID, asks if you agree to marry the other, he pronouces you man and wife. About 3 to 4 minutes tops, If you have ring, you can exchange rings, he normally just says do the rings.