Lightroom uses ProPhoto RGB, the monitor in Adobe RGB. Does this mean that the colors I'm seeing through the monitor when editing is being translated to Adobe RGB?

Yes. But not quite. Your monitor is (likely) not exactly AdobeRGB; it has its own colour space which for your Asus should be close to AdobeRGB. Now, whenever you work with a colour-managed application, colours are being translated - at least - from the image space to the monitor space.

In case of Lightroom (which I don't use, but it should be similar to CameraRaw, which I do), there is another intermediate translation to ProPhoto RGB. When one does many complex calculations, it's just more convenient to work in a fixed known reference system. Obviously, in order not to compromise the end result, this intermediate reference system must encompass both the input and output, and ProPhoto RGB is the only common standard that (more or less) ensures it.

This intermediate stage should really be transparent; it mostly facilitates internal algorithms, but fundamentally it doesn't change much.

If the above is correct, and I eventually export my images as JPGs (sRGB), does this mean that the JPG version could end up looking different than what I was previously editing? It would be worse if the exported JPG would be viewed through an app that doesn't manage color. Is this correct?

The worst thing that can happen is if you export the photo in a non-sRGB space and then view it on a regular non-colour-managed monitor. All colours will be affected then. When you use sRGB, you just have better chances that 90% of people who use such regular monitors will see something reasonable.

For yourself, if you just export as sRGB and view it, overall it will look similar, but the most saturated colours that fall beyond sRGB but which you could still see on your wide-gamut monitor will be clipped. (The way they are clipped depends on the rendering intent you selected during the export).

For this reason, when you edit the photo, you already should select the output (export) colour space. I don't know how Lightroom handles this, but in CameraRaw you have to do this. This serves two purposes:

• It 'proofs' the colours as they will appear on the exported image. If you select sRGB, you won't see very saturated colours during editing.
• The histogram will show you the actual output with all the clipping that will happen during export. Or conversely, if you want to export in a wider ProPhoto or something, you'll see the truth in the histogram but not visually (in the most saturated areas). This may affect your editing decisions.

Alternatively, if I exported it with AdobeRGB, then the during edit and after export version would be the same?

If your software proofs the colours during editing like I described above, it will always look the same. I'm not sure about Lightroom, but I don't like the idea of editing without a reference to an export profile. The ProPhoto and AdobeRGB histograms look dramatically different.

If the software doesn't proof, then sticking to AdobeRGB (in your case) will give you the most consistent results. But again, in a colour-managed workflow, it's all a matter of the most saturated colours: you should only notice a difference there, if any.

This JPG's colors would be viewed correctly even with an app that doesn't manage color (since the file and monitor is using AdobeRGB)?

On your nearly-AdobeRGB monitor, yes. But you (and everyone) should use colour managed apps anyway.