I want to follow-up and close the loop on this issue as it is now solved.
Tcpdump and the iptables logging rules that I inserted on the gateway box did not help with diagnosing this, as the logged data was not giving me any additional clues (it is obvious why this didn't help when you see the solution). So I went back to the beginning and reviewed all my network configuration and did some more reading on NAT. This article www.netfilter.org/documentation/HOWTO/NAT-HOWTO.txt
is what pointed me in the right direction and helped me find my configuration error. The article is a little old, but it is well written in an easily understandable style and is still relevant. I would recommend it to anyone trying to understand iptable rules for NAT.
The two errors I had were that you don't need a 1-1 NAT rule for the gateway box itself, as it is the destination point for packets coming into the box so INPUT rules are what is relevant there. Thanks Nick for pointing that one out. The second error was that the boxes behind the gateway had the wrong default gateway address. I had installed the gateway box into my existing working network between the cable modem and the switch. The cable modem was still referenced as the default gateway on the few boxes behind the gateway that I was trying to access. So this resulted in it appearing that packets from my remote access attempts to the boxes behind the gateway were being silently dropped. Whereas what was really happening is that packets were getting to the boxes, but the replies from the boxes were getting mangled or dropped because they were skipping past the gateway box that would do the 1-1 NATing going back out, and instead going to the cable modem box as their gateway.
I now have full administrative access to my boxes via the 1-1 NATing, and am working on further securing the access to them from potential hackers.