Sorry, my knowledge of dnsmasq is somewhat limited, so not sure how much I will be able to help at this point. A couple of suggestions though:
1) You /etc/hosts file still is not right. Go back and look at the Split DNS wiki page. Your file should really only have two lines. The localhost / loopback line, which looks OK in your file, and then one other line which should contain the IP address of your server, followed by the FQDN of the server, and then the host name (unqualified). So, if your server is called "mail" then the line should be "10.3.0.99 mail.axiomcomputerservices.com mail"; if the server is called "zimbrahost" then the line should be "10.3.0.99 zimbrahost.axiomcomputerservices.com zimbrahost". You want one or the other, not both lines in the file. If you are trying to setup aliases or something like that it should be done in the DNS setup, not here.
2) While your resolv.conf file looks OK to me, you really need to understand how it works. When you have multiple "nameserver" lines it will only go through the progression of servers if, and only if, the servers are not responding at all to DNS requests. So, in your case assuming the DNS service is running on the local system (127.0.0.1) all name resolution on this, and any other server that is using this as its DNS server, will be done on this server. That means if the service is running, but not configured properly the name resolution will just fail; it will not drop down to your 2nd or 3rd name-server. It will only go to the next server if the DNS service is not running at all. In the Split DNS scenario, at least to me, it doesn't make any sense to have external IP's in there.
3) I don't know if you have just altered the "MX record" error that you posted, but it says "zimbrahost.mydomain.com", and it seems like you have answered a previous setup question incorrectly or there is some other configuration error on the server. I think you want your email domain to be "axiomcomputerservices.com" and your DNS server needs to answer the "get MX record for axiomcomputerservices.com" request properly.
4) As I said I don't really know dnsmasq that well, but your config file does look similar to what is posted in the wiki. Have you verified that this service is actually running, and if not checked the system log for any relevant errors?
As Phoenix has already suggested, I would recommend to go back through the Split DNS wiki page and check out the "Verify" section. If you are still having problems post *exactly* the answers you get to each check.