Show the address on button

Unfortunately no.
The [ethpress_account] short code can be used to display the current user’s address logged with.

I see. And is there a way to autorefresh the page after linking the wallet to the website?

Good catch! I’ll add it to my roadmap.

You can use this hook as a workaround for now:

/**
 * Fires after every user account linking success.
 *
 * @since 1.5.0
 *
 * @param WP_User|WP_Error $user WP_User on success, WP_Error on failure.
 * @param (string|false) $provider One of 'metamask', 'walletconnect', false.
 */
do_action( 'ethpress_linked', $user, $provider );