Firstly there's not a whole lot of reason to strip exif data. Mostly it relates to the camera, focal length and white balance (etc etc), and the only identifier is GPS data, which if you have the gps turned off in your phone it's a non-issue.
If it is an issue though, then if you have windows you can do it using this button. Just select all you photos in explorer and right click, then go to properties. There's probably an option in OSX for this too though.
You can do it using software (lightroom and photoshop both can do this - just export without checking the 'include metadata' (or something) box. if you don't have those, then you can use an app like exiftool, which is a command line tool but you can get a GUI for it as well - it's pretty fantastic.
That option can strip all the data from a file though which may be overkill.
If all you want to get rid of is the GPS data though you could just retake the photo with that option turned off in your phone app.
I'm afraid I don't know / don't use photoshop. I'm on linux, so I just do exiftool -comment="This is a metadata comment" Sample.gif. ExifTool also available for windows but there are a couple steps to set it up http://u88.n24.queensu.ca/~bogdan/
True, and I have thought of this but I haven't learnt how to use exiftool (it seems intimidating to be honest). All the apps I tried which are powered by it seem to lack the functionality you describe (use a specified exif attribute to replace other specified attributes).
I've been pointed toward exiftoolgui, and it seems like that will be able to reassign exif data with ease, but I ran into some errors when trying it out. There's also a custom script I can use (python) recommended by a person on Advanced Renamer that will accomplish the main task, so I'm currently juggling between the 2.
It will depend on the software you're using. Windows Photo Viewer is usually the default, but it could be set to MS Paint or another app. And I think Win10 might auto-rotate when you open in Photo Viewer. But it sounds like the app you're using to view the photo isn't writing to the Orientation field, and the site you're uploading to is reading the unaltered data.
You can view the Orientation data in Windows Explorer by adding a column; right click in the column headers (next to name), click "more", scroll down until you find "Orientation" and check the box. Now you should see a column that shows the orientation of the images.
Alternatively, you can download ExifTool
http://www.sno.phy.queensu.ca/~phil/exiftool/
You can use this tool to view the EXIF data
exiftool.exe {filename}
and if your program isn't changing the Orientation, you can use ExifTool to reset the Orientation field.
exiftool.exe -Orientation=Normal {filename}
If you're not comfortable with command line, there's a third party GUI which makes it easier for Windows users. http://u88.n24.queensu.ca/~bogdan/