Conduct the following experiment

Yes, "do the following experiment" sounds more natural. IMO, "Conduct" conveys precision and is appropriate for more formal writing.

I'll say, run the following experiment.

Perform, conduct, execute, the experiment, etc... You generally would set up an experiment, after making a hypothesis. You could make up an experiment, and then execute I suppose. It's a noun and a verb, so you could experiment with making experiments, but making a following experiment would then be making a made thing, so I don't think that would be correct.