Keep in mind that if viewing on the web, on sRGB is supported.
No, TN panels cannot display the full range of any of the colour spaces. To give you an idea, most TN panels display 6 bits per colour natively while JPEGs do 8-bits.
Sure they can. 6 bits vs. 8 bits has nothing to do with the color space spanned. That just tells you the "resolution" that monitor will have within its gamut. There are plenty of TN wide gamut monitors that span all or near all the aRGB gamut.
Which you pick largely depends on your purpose. Typically I would just select sRGB. If I want to print, I'll start with the RAW file and convert it to aRGB space. For image saving purposes I'll save in sRGB. The difference is relatively minor in most cases and can be previewed in photoshop if you are curious.