Step 1: Add the names
Everyone who joins is added. On a paid plan you can type your own list.

A random name generator for the big screen. Let everyone join, then let chance pick.
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Yes. The Random group maker splits everyone into groups on a paid plan, and the random team generator does the same job.
Yes. Spin a prize wheel or run a raffle duck race, so everyone sees the winner at the same moment.
Use the random team generator and set the number of teams to half the number of people: 24 people make 12 random pairs.