There is no requirement in SMTP for a domain to have an A record. While valid, and normal for MX domains, using the domain A record to identify the MX is not recommended. From your description you have a separate MX domain, so you don't need an A records for you domain.
It is not uncommon to have the A record(s) return the address of one or more of your web servers. This allows people to find your web presence more easily.
To be set up properly, your MX needs a fixed IP adddress, an A record. and a PTR record. The A and PTR records should contain the identity of the other. Normally, you will need your ISP (or IP address provider) to configure the PTR record for you. Your mail server should identify itself with the name in the PTR record.
example.com. IN MX mail.example.com.
mail.example.com IN A 22.214.171.124
10.2.0.192.in-addr.arpa. PTR mail.example.com.
You may want to use SPF records for the domain and MX. TXT records were originally used and some tools only look for them. Creating both SPF and TXT record should minimize lookups required.
example.com. IN TXT "v=spf1 mx -all"
example.com. IN SPF "v=spf1 mx -all"
mail.example.com IN TXT "v=spf1 -all"
mail.examle.com IN SPF "v=spf1 -all"