Ok, wait and stop.
Two parts here, first the important one.
As almost 95% of these cases... You do not mention at all the color profile you are using or need.
Corel X7 does a decent job when converting color modes, but you need to have it configured right.
Find out what type of paper you are going to use. Mainly coated or uncoated. I am guessing coated.
Define the region of the world and the specific profile the press is using. I am guessing Europe here, but one common problem is that people recommend Fogra profiles that use a TAC of more than 300%... and then they decide they need a 300% top.
I'm sorry, my Corel is in Spanish, but here is a screen capture:
- Go to Tools > Color > Default preset?

And on the screen choose one based on the information of the publisher.

Second part
But the specific problem you are describing is that you are testing your file on a heck of different programs that manage the color profiles differently. Some simply ignore them.
The last image you posted is simply discarding all the CMYK previsualization and colors profile because some web browsers can NOT handle a CMYK file correctly, so do not worry. The same on XnView.
The first image is an RGB file before the conversion it is probably simply using the RGB profile or not using any at all, which on an RGB file is probably not a big deal.
The second is one screen capture of the program actually simulating the color conversion. That is what a pro program do, give you a representation of the colors simulating how they will change on print. If you did embed the color profiles that will be fine.
The press is not going to open the file on a viewer, or a web browser, they will work within a workflow of programs that consistently handle the embedded color profiles and CMYK values. This is why the first part is the important one. The ideal thing would be that your monitor has some minimum calibration.
Let me spam you with an ugly old webpage of mine. https://otake.com.mx/ColorCalibration/ The most important part is the gamma. The modern standard is 2.2.
Edited. Part 3
You need to prepare two different PDF files, but not the color issue. Adobe reader is one of those programs that make a good representation of a CMYK file inside it. So you could see the final CMYK PDF file prepared from Corel and use adobe viewer to see it.
But in reality, you need to prepare a separate PDF to be viewed online for the file size.
When exporting a PDF File for prepress, the file is normally quite big for two reasons.
- The Raster images need to be at high resolution, let's say the typical 300PPI
- They need to be CMYK and with minimal compression, this is using only ZIP compression. Even using High-quality JPG compression (which I only recommend in case you know what you are doing) a CMYK file does not compress well.
For a good PDF to be viewed online it is a good approach to prepare a reduced version of the PDF, with images on RGB mode, resampled down, let's say to 100PPI which makes them 9 times lighter and with some good JPG compression to make them, not perfect but easy to handle. This will produce a PDF file lets say 50 times lighter than the one made for print.
Part 4
how can I be confident that it will be correct in the final version?
By maintaining al the Color Management workflow all the way. I will expand it from the photoshoot itself but highlight the ones relevant to your part.
A good illumination with lights with good color consistency (No cheap fluorescent lights or cheap led lights for example)
Good exposition and white balance on the camera. Images saved with a defined color profile.
3. Consistent management of the files, no discarding of the embedded color profile.
4. If for some reason the images have not an embedded color profile, you need to decide at this point to embed one. For print, a commonly used is Adobe 1998, for web-only sRGB, for both, stay with the Adobe one.
5. If you need to edit them, use a calibrated monitor and a program that actually previews this (Corel Draw and Corel Photopaint are a good option with the settings as I showed you)
6. Work in RGB as long as you can with the color simulation turned on, leave the conversion just as the last step.
7. Send a correctly prepared file with the profiles embedded.
Ask for feedback of the printer.
Ask for a digital print well-calibrated. This is easier if you live nearby, but sometimes they can send a printed simulation overseas... Ask.
But the real thing is standing in the print room when the print is actually happening, and grabbing the first fresh samples. This sounds silly on these modern days but for some crucial projects, this could be imperative the first time you work with a new provider.
With a well-calibrated workflow, you can stop at point 8 and be confident that things will go fine.
An alternative tho steps 3-7 is to leave the color handling by a studio.