I have bought a domain which should only be used for email addresses. The SMTP server has the name mail.mydomain.com. When checking the domain configuration using some web service like mxtoolbox.com, it complains that the website mydomain.com is not available. Is it really necessary to have a website if using a domain for mails only? If it is not necessary, could mails be rejected by recipients as spam if I do not have a working website in place? I.e. is it recommended to have a website? Is there a standard (like RFC) where this topic is discussed?
No. You don't need a website for an email address and having one does very little if anything for your (email sending) reputation - certainly email won't be rejected just because you don't have a website.
The only thing I can think of that might be a bit relevant (again not very important at all) - if you dont have an MX record set, a mail server is supposed to fallback to trying to deliver email to the A record (i.e. a DNS entry that is now more commonly associated with websites). Again, this is little more than a relic of the early days of the net and not generally important.
The only requirement for an email server is to be reachable. In that sense the domain must be hosted somewhere, and there are only two options: Your computer email server or a web hosting service. In that respect, your own website is not strictly necessary.
It seems that in your case the vendor of your domain name does this service for you, by hosting your domain and supplying the required DNS records and services.
In that respect, you may ignore the warning messages from
as long as your mail is functioning.
You should instead verify your email server by one of the free Email Checkers found on the internet. There are multiple authentication standards that your email server must obey if you wish your emails not to be branded as spam. For example DMARC. It is important that your email hosting service support all of them.
As the others have pointed out, it is not technically neccessary to have a proper website attached to it.
However, some mail providers (like the German Telekom) may sort your mail into spam or outright reject it if they do not find something resembling an imprint or a link to one on your main domain. (Source, in German: https://postmaster.t-online.de/#t3.2)
You do not need a webhosting attached to the domain, as the MX record can point to any mail provider by itself.
If the mail is hosted on the same server that normally the website would be hosted too, it is likely not possible to get a certificate running, and thus, you lack SSL/TLS support. With increasing blocking of insecure mail traffic, that is the only thing that could become an issue.
But again, its not mandatory to run email.
That said, although mxtoolbox in the name implies it is for email, mxtoolbox does far more. Its likely that you did a DNS check and that it complained about an A record, not the MX record.
Then again, many webhosting providers offer both mail and webspace. The mail is then stored on the webspace, and you can even choose to not make the domain point to the website, turning it into a mail-only solution. But technically, that is still a webhosting.
So if you really only have a domain, but no space behind it, you cannot expect mail to be working. You need a mailserver, either from a dedicated email hosting service or by getting webspace.