WordPress replaces my permalinks!

What permalink structure are you using?

I’m using a custom structure: /%postname%/

Have you checked whether mod_rewrite is running properly on the server?
What’s in your WP root .htaccess file?

phpinfo says:
Loaded Modules core mod_win32 mpm_winnt http_core mod_so mod_access mod_actions mod_alias mod_asis mod_auth mod_autoindex mod_cgi mod_dir mod_env mod_expires mod_headers mod_imap mod_include mod_isapi mod_log_config mod_mime mod_negotiation mod_rewrite mod_setenvif mod_userdir mod_vhost_alias mod_php5 mod_dosevasive22

Also, everything used to work fine (for at least 4 months) until a few weeks ago. No new plugins were installed or anything like that.

The pretty permalionks work great – i mean, when i type https://www.ecovisvesiir.ee/sundmused/ or https://www.ecovisvesiir.ee/?page_id=483 on my browser, in both cases i’m taken to https://www.ecovisvesiir.ee/sundmused/.
The problem is that when I create a link isnide my post or page, and put it as https://www.ecovisvesiir.ee/sundmused, WordPress then replaces it with https://www.ecovisvesiir.ee/?page_id=483 when I save…

There may be a plugin conflict. Try deactivating all plugins. If that resolves the issue, reactivate each one individually until you find the cause.