install / uninstall, update, introducing the key, nothing helps. What should I do?

my site homerobot.com.ua

Hi,
Make sure that you have correct Download ID set in JoomSEF's Configuration on the Registration tab.
Then open the Upgrade page in JoomSEF and check if your registration information is correctly recognized by our server. If it is, click the Reinstall from ARTIO server button (or Online Migrate to Paid version). After that, the paid version should be installed automatically and ARTIO link should disappear.
Also make sure that in JoomSEF's Control panel in the SEF Status tab everything is enabled.

and by reinstalling from the server, it's not going to wipe out all the work I've done on the component so far right?

That's correct, if you don't uninstall the component manually, reinstall will not remove any of your SEF URLs or configuration. But of course having a recent backup of your website (with database) using eg. Akeeba Backup component is always recommended before updating any component.

I followed the instructions and it all worked. Had the same problem.

Thanks!