FCP doesn't always get the frame rate right when making a new project,

The clip frame rate is whatever it was recorded as. You can of course drop a 29.97fps clip on a 23.98 project, or vice versa.

(By the way, Moiré patterns should have nothing to do with frame rates. More on this later.)

When you create a new project, you have the option to use Automatic Settings, or Custom Settings.

In the former case, the project should take on the properties of the first clip that drop into it.

In the latter case, you choose what you want; it might be 1920x1080 at 29.97fps or whatever.

It should remain just as you chose.

So, let's see if you we are on the same page:

1) Make a new project; if necessary, click Custom Settings; choose the desired frame rate and frame size

2) Press Command-J and see that in the Inspector the settings are ok

3) Paste your clips into this project

4) Repeat step 2; has anything changed? It shouldn't...

You are exporting your work as 1080 29.97i Interlaced:

Is that your intention? Interlaced video does not display correctly on computer monitors, it needs to be displayed on a TV monitor that supports interlaced video (most TVs) to be displayed correctly.

MtD

Regarding Moiré patterns:

It would help a lot if you could post two screenshots, one from the viewer and one from the exported movie, to illustrate the problem.

We also need to know what your export settings are.

One common technique to avoid this problem is to add just a little bit of noise to the image.

This prevents compression, to some extent, from causing the issue.

Again, some screenshots might help.

Even better, if you could take a short original clip that is affected, and post it - you could send it to yourself using wetransfer and post the link here. We could do some testing.

Are you referring to the jagged lines in the panel in the background of the speaker?

Could you post a screenshot of the Viewer in FCP? Does it show the same problem?

Does it show the same problem when you open the original clip in Quicktime Player?

Meg made a good point about exporting as interlaced?

Is the footage interlaced or not? Is the project set as interlaced or not?

<rant>I wish that interlaced video and fractional frate rates would die already; they should not

be used in digital video in the twenty-first century </rant>