Is an MX-record required on a domain to send emails only?

No and yes.

By standard, your sender domain is not required to have an MX record. RFC 5321 makes an MX record optional.

However, an MX record is somewhat of a standard today. Due to spamming practices, a number of receiving servers may reject your messages when there's no MX record for the sender's domain. Since it is no big deal to set up an MX record it is highly recommended to do so for serious traffic.

As @MichaelHampton has pointed out, you need to be able to accept messages for the sender's domain in any case - including the postmaster@ and abuse@ aliases as per RFC 2142 - to avoid being categorized as spam.

No. A MX record only determines which mail server mail to a user at that domain/subdomain should be ultimately delivered to. Without an MX record, it will attempt to deliver to the actual hostname in the email address, so keeping @MichaelHampton 's comment to your answer in mind you'll still need to be able to receive bounces, etc. but as long as you accept mail (even if routing to /dev/null) you should be OK on that end.

No DNS records are required to send email, however, not having appropriate mail-related DNS records for your domain is likely to result in getting discarded by spam filters. Your shop.mydomain.com server should be probably be treated as a host authorized to send mail rather than as a child domain.

I know of at least one service that will up the spam probability score if an MX record is not present for the domain. Having other DNS entries, like valid PTR records for mail servers and records for Sender Policy Framework (SPF) and DomainKeys Identified Mail (DKIM) for the domain will also help.

There is a good article by rackAID, 3 DNS Records Every Email Marketer Must Know, that outlines the basics of DNS interaction with spam filtering.