email server - Reverse DNS is not a valid Hostname error from MXToolBox

I can't send email messages from my server. However, I can receive email messages. I set up my server the exact same way I did many times which I've proven it works and it's pretty straightforward and consist of: Create a DigitalOcean Droplet (Debian 7.0) + Set PTR record through renaming Droplet to a FQDN + Create Private NameServers in my Domain Registrar and point my domain(s) to my NameServers + Installing VestaCP (child-ns). From here, I simply add a web through VestaCP and the DNS records are added automatically.

I am not sure where I have gone wrong in my configuration but I am getting the error "Reverse DNS is not a valid Hostname" for my domain izanami.co. I set my DigitalOcean Droplet PTR record to izanami.co and all of the tests I have conducted using dig and host seem to return the correct information.

dig -x 188.166.32.113

; <<>> DiG 9.8.4-rpz2+rl005.12-P1 <<>> -x 188.166.32.113
;; global options: +cmd
;; Got answer:

;; ->>HEADER<<- opcode: QUERY, status: NOERROR, id: 47119
;; flags: qr rd ra; QUERY: 1, ANSWER: 1, AUTHORITY: 0, ADDITIONAL: 0

;; QUESTION SECTION:
;113.32.166.188.in-addr.arpa.   IN  PTR

;; ANSWER SECTION:
113.32.166.188.in-addr.arpa. 1283 IN    PTR izanami.co.

;; Query time: 9 msec

;; SERVER: 8.8.4.4#53(8.8.4.4)
;; WHEN: Fri Aug  7 15:06:39 2015
;; MSG SIZE  rcvd: 69

hostname -f

izanami.co

host izanami.co

izanami.co has address 188.166.32.113
izanami.co mail is handled by 10 mail.izanami.co.

host 188.166.32.113

root@izanami:~# host 188.166.32.113
113.32.166.188.in-addr.arpa domain name pointer izanami.co.

dig +short ptr 188.166.32.113.in-addr.arpa

113x32x166x188.ap113.ftth.ucom.ne.jp.

MXToolBox Transcript

Connecting to 188.166.32.113


220 izanami.co ESMTP Exim 4.80 Fri, 07 Aug 2015 14:20:52 +0200 [5750 ms]
EHLO PWS3.mxtoolbox.com
250-izanami.co Hello pws3.mxtoolbox.com [64.20.227.134]
250-SIZE 52428800
250-8BITMIME
250-PIPELINING
250-AUTH PLAIN LOGIN
250-STARTTLS
250 HELP [719 ms]

MAIL FROM:
250 OK [750 ms]
RCPT TO:
550 relay not permitted [1141 ms]

PWS3v2 9719ms

I tried to send an email message and this is the log output in /var/log/exim4/mainlog

2015-08-07 14:58:07 1ZNh1m-00025M-ON == web-3ebnda@mail-tester.com  R=dnslookup T=remote_smtp defer (-53): retry time not reached for any host

Update:

A few more lines from my /var/log/exim4/mainlog. As you can see, an email message which is supposed to be (auto)forwarded to my Gmail account doesn't succeed.

2015-08-07 17:36:12 H=pws3.mxtoolbox.com [64.20.227.134] F= rejected RCPT : relay not permitted
2015-08-07 17:49:24 SMTP protocol synchronization error (input sent without waiting for greeting): rejected connection from H=128.red-79-157-253.dynamicip.rima-tde.net [79.157.253.128] input="^\r\n"
2015-08-07 17:51:47 1ZNjvu-0002TQ-Vb DKIM: d=google.com s=20120113 c=relaxed/relaxed a=rsa-sha256 [verification succeeded]

2015-08-07 17:51:47 1ZNjvu-0002TQ-Vb <= send-as-noreply@google.com H=mail-la0-f52.google.com [209.85.215.52] P=esmtps X=TLS1.2:RSA_ARCFOUR_SHA1:128 S=3356 id=CANY4h_3COHYMBQt1RA0QVgEc-LRv6ErtQPh68njd=dqPp_G=VA@mail.gmail.com
2015-08-07 17:51:47 1ZNjvu-0002TQ-Vb => example  R=localuser T=local_delivery
2015-08-07 17:51:47 1ZNjvu-0002TQ-Vb alt2.gmail-smtp-in.l.google.com [2607:f8b0:400c:c06::1a] Network is unreachable