Do mail servers and/or spam filters care about a domain's root A record?

You don't need an A record at the apex of your domain for e-mail to work.

However if you don't have any MX records, any A (or AAAA) record found will be treated as if that host was an MX with preference 0.

See s5.1 of RFC 5321.

From RFC 5321, Section 5.1:

When a domain name associated with an MX RR is looked up and the
associated data field obtained, the data field of that response MUST
contain a domain name. That domain name, when queried, MUST return
at least one address record (e.g., A or AAAA RR) that gives the IP
address of the SMTP server to which the message should be directed.
Any other response, specifically including a value that will return a CNAME record when queried, lies outside the scope of this Standard.

What this means is that your MX record HAS to point to an A (or AAAA) record to be valid. If the A record that you want to remove is where your MX is pointed to, then you can't remove it (and expect to get your email)

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   192.0.3.10
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"