Questions about Sigma TC-2001 to 2.0x Nikon

No

It already works. Whether the autofocus still works, however, is questionable.

According to Sigma's compatibility table, yes: https://www.sigma-foto.de/fileadmin/relaunch/products/Telekonverter/Kompatibilita%CC%88tstabelle_12-2018.pdf https://www.sigma-foto.de/objektive/tc-2001/technische-daten/

This product is compatible with the Sigma 150-600mm F5-6.3 DG OS HSM lens, as well as with Nikon F-Mount cameras, so it should be compatible with your setup.

No, it works exclusively with Sigma 150-600 Contemporary / - Sport and a few other Sigma objectives. There is a table published on the according Sigma webpage, that shows with which objective it's compatible and whether AF works: https://www.sigma-global.com/de/lenses/cas/product/popups/tele_converter/

No. Teleconverters are intended for professional lenses with continuous apertures of 2.8. and max 4. If you need extra focal length I recommend e.g. a Tamron or Sigma 150-600mm... everything else makes no sense and is even mechanically often not possible. If it is mechanically possible then you lose autofocus and image stabiliser. To achieve a long focal length I recommend a DX body (1.5x crop) and a 150-600mm. This will give you 225-900mm.

With my Nikon 7100 and the Sigma 150-600 Contemporary, the AF unfortunately only worked up to about 450mm, after that you have to focus from Manuel. The multiple measuring fields don't work either, and so I have already returned the converter... :-(