Ubuntu exim4 - Config setup and spam filters - email

Question

I've recently setup my Ubuntu web server with exim4 so my PHP website applications can send email such as "thank you" and "confirmation" notices.
I've got it setup and working such that I can send email to gmail, Yahoo! and my work address. However, my work email gets caught up in our spam filter. I'm new to setting up mail servers so I'm not sure what I might need to look for in making this mail server more trusted, while keeping is secure.
Here are some details:
Server is NATed behind a firewall.
Firewall has port 25 open for outgoing SMTP traffic (from server to anywhere).
Server is virtual hosting a couple different of our websites
The server is running the following exim4 config:
dc_eximconfig_configtype='internet'
dc_other_hostnames='web-serv.example1.com;example2.com'
dc_local_interfacees='127.0.0.1'
dc_readhost=''
dc_relay_domains=''
dc_minimaldns='false'
dc_relay_nets='' dc_smarthost=''
CFILEMODE='644'
dc_use_split_config='false'
Questions:
Do I need to open port 25 to incoming SMTP mail (anywhere to server)? I wonder if other mail servers need to talk to my mail server to verify itself, in a sort of handshake attempt.
I have not created any MX records primarily because the server has different websites on it the mail server should send mail for all the websites. Do I need to pick/create a domain address and create MX records for it?
One thing of note is that the mail headers look like this:
Return-Path: <www-data#example2.com>
Received: from web-serv.example1.com ([Firewall public IP Address])
Received-SPF: neutral (google.com: [Firewall public IP Address] is neither permitted nor denied by best guess record for domain of www-data#example2.com)
"web-serv" is the host name of the server, such that you get this if you type it into the command line:
$ hostname
web-serv
and "www-data" is the account name for the Apache2 server that Ubuntu gave it as default.
Any other general advice would be appreciated. It's all new to me.
Cheers!

Answer 1

One item of note, since I posted this question time time ago (almost 10 months) is that I found out the biggest issue I had was with setting up the DNS for reverse DNS on our hosting providers side of things.
In other words, our hosting provider (the people who give us our IP address and manage our hardware) had to enter a record to match my server(s) hostname to whatever IP address it used.
There's a specific name for this. I believe it's a "PTR" record but the name escapes me at the moment, but you basically tell them "my server hostname is ..." and they do a quick update to the DNS for reverse DNS purposes.
When I asked this question, we had a different hosting provider who didn't really help explain this to me, and after switching providers, I got to talk to someone who was happy to help me understand that side of the equation.
And as I understand it, this is setup by the people who assign you the IP addresses. But there's probably more to it than that.
Once I got that setup properly, email had no problem getting through the spam filters and Gmail/Yahoo showed SPF as "passed". It was showing neutral before.
Our company email was set to drop any email that would not resolve reverse DNS, which is why I could not even receive the email or find it in the spam filter. Of course, that situation would be dependent on the company and what email policy and software they're using to manage spam. Some might just drop all email that does not reverse DNS and some might dump it in to spam filters instead.
Hope that might help some people with similar issues.
Cheers!