Are you facing this issue only on Wi-Fi connection  or even with direct connection? 

Since you have done checking with your service provider multiple time you can reach out to Arris.com/consumersupport if your modem is under warranty.

We are experiencing dropped connections, both on Wi-Fi and direct connection. When I went into the testing mode, it says packets received but not sent. I sent multiple emails to: consumersupport@vantiva.com as directed,  with copy of purchase receipt, which is the address I was given, but have not received any replies. Case #: 2404-06006. I have repeatedly asked for an acknowledgement that the email has been received, but.....nothing I would really appreciate a reply as to my warranty and what can be done. It's only getting worse.

Since you are experiencing intermittent internet connections on both wired and wireless connections, I suggest you to check the cable signals to make sure all signals are within range.

http://arris.force.com/consumers/articles/General_FAQs/G34-G36-Cable-Signal-Levels

This was all done numerous times. Everything regarding the cables/ service provider, etc is working correctly. The only thing NOT working is the modem router.

What I need to know is: is the modem router under warranty as I purchased this in July of last year?

Since you tried all the possible troubleshooting, I suggest reaching out to www.arris.com/consumersupport for further assistance.

I HAVE reached out to consumersupport@vantiva.com, which is where I was sent 2 weeks ago. Absolutely no reply. I've tried everything. Does Arris not care about their products or their customers? 

The email address that you mentioned is not a valid email address. I suggest reaching out to www.arris.com/consumersupport to get a valid email address.