Can I find somebody's else coordinates through images without any plugin/mod and them knowing?

I don't think it's possible to do without a mod or plugin.

However If you are in bedrock you can use a map to see another players location (not there coordinates specifically) and you clone the players map on java

However you will need mods for the best results.

You can also use achievements or signs that a person is in the area such as a nether portal, unnatural holes in the ground ect.

Yes, you can. The way to do this is to know the seed of the world(which you have) and compare the seed's terrain to the terrain in the images. If you are playing on an older server(like 2b2t), you should probably try looking for matches of the terrain in older versions of Minecraft with different world generation before trying it in the current version. This would be able to give you the coordinates to the base. If you are a Java programmer, you can also write a program to help automate this process. In fact, this is how some base hunters on 2b2t track down bases in images, and also why many 2b2t players advice you to never take a screenshot of your base, as they can use the method I stated above to find your base and grief it/leak its coordinates. I hope this helped. Thank you.