OK, so in traditional fashion, after typing something out here, I think I found my own solution…
Problem 1. The unresponsive web interface for the backup turned out to be the WebConfigurator Lockout rule getting in my way. I found this by going to Status -> System Logs -> Firewall. I looked at the blocked log entry coming from my source IP to the firewall's destination IP. I click on the Red Block symbol on the left, and it tells you what rule blocked it. To solve the issue, I removed all of my allow rules from the NAT and Rules tables allowing my external IP in to this firewall, and then I rebooted the firewall under Diagnostics -> Reboot -> Yes. Once the firewall rebooted, I went into the rules table, created a new rule:
Protocol: TCP
Source: {MySourceIP} (I used an alias)
SourcePort: Any
Destination: WAN Address
DestinationPort: {MyAdminPort}
Gateway: Any
Queue: None
Schedule: None
And voila, my linux server can access the firewall Admin port again.
Problem 2: Login failing with wget. This turned out to be stupid simple. Linux doesn't like the apostrophes as delimiters for the POST data. It wants quotes. Rough, huh?
Good luck, and hope this solves it for someone/everyone.