Nexus Mods and Vortex use a totally different game structure (Basically anything Bethesda built), so you'll still be downloading them manually. I don't even know if Vortex can run Kotor nor would I advise trying, since it uses a totally different engine and loading process.
If you downloaded 7Zip and installed it, you should be able to Right-Click the zipped file (Rar or Zip, it doesn't matter) and then "Extract here". You could also use "Extract to [name of the file]" which will create a new folder for the download to be in, but it isn't strictly necessary. (Your folders will look pretty messy though)
Please note that TSLPatcher is not designed nor is it advisable to run it from inside of a zipped folder. You will want to extract it before continuing past this step.
TSLPatcher is a utility that runs like any other EXE; you double click it and follow the on-screen instructions and then select the folder your game is installed in. This varies, but for me it is installed in my C : Drive under Lucas Games and then "SWKOTOR2". Bear in mind that TSLPatcher changes the main files inside of the Modules folder and other main files, so it's always good to use the Backup folder created beside the TSLPatchData folder if anything goes wrong.
For "loose" files such as textures and some UTCs (Always follow the instructions on the Readme to prevent installing things in the wrong place), you can extract the files and then move them to your Override folder which is basically a special folder that overwrites the game files with whatever is in it. There are also some mods that require you to move the files to other locations like "StreamMusic" or "Movies", but they are generally rare and as long as you follow the instructions in the Readme, you should be fine.
If you have any issues, then we will see if we can figure them out.