WP24

Da das ein kostenpflichtiges Theme ist kann ich das nicht testen.
Können Sie einen Link zu der Seite posten?

I don’t know what you exactly mean with dropdown. But since you are using an other theme it’s difficult to change the design to your example.
You could use a button instead of links, to do so change the WooCommerce settings from “[link]buy now[/link] for [price]” to “[button]buy now[/button] for [price]”.
For other stylings you would need to put custom CSS, e.g. to add a gray line do:

.wp24-dc .table-cell {
  border-bottom: 1px solid lightgray;
}

Does the page reload after hitting the save button? The page should scroll to top then. Maybe you need to make your browser window smaller to test.

Is anything displayed or is the check form completely missing on the page you put the shortcode?
Could you post a link to the page where the problem occurs?

In that case you need to style the textfield and button via CSS. Just to give you an idea:

#dc-domain-1 {
	height: 75px;
	margin-right: 0;
	border-bottom-left-radius: 50px;
	border-top-left-radius: 50px;
}
#dc-submit-1 {
	height: 75px;
	background-color: orange;
	border-bottom-right-radius: 50px;
	border-top-right-radius: 50px;
}

The CSS needs to modified depending on the theme you are using.

You can change the width of the domain name field in the general settings of the plugin.

There are also several recaptcha settings where you could also enter your keys. Which problems exactly do you have, to enable it?

Port 43 needs to be opened by your hosting provider, otherwise the whois queries won’t work.
You can find the requirements for whois queries in the plugins settings under system info.