What Happens When SYN and FIN Flags in TCP Headers Are Both Set to 1?

Does that make any sense?

No.

The FIN is send because the sender decided that it wants to close the connection. Even if you would change the recipient that it will ignore the FIN the sender side of the application will still consider the connection closed and not send or receive any more data on it.

Just compare it to a phone call. If the other end decided to hang up the phone and no longer talk to you it does not matter if you are trying to ignore this, the other end still will not talk to you any more.

What are my options?

This must be fixed inside the application. For example the application could reuse the existing TCP connection like done with HTTP keep-alive.

You are probably developing the new way of hidden tunneling )

As to the fields of IP, TCP, ICMP headers - you can't be absolutely sure that they will be untouched traveling the network. It mostly depends of source and destination network topology, security measures and network equipment or software. Even, the direction of traffic matters. It much differs when traffic flows for ex. from home network to your VDS server or from corporate PC to google server behind load balancer etc..

• IP destination address as well as TCP/UDP destination port are likely untouched unless destination is behind the NAT or load balancer
• TCP SEQ|ACK can be sometimes altered by some firewalls (Cisco ASA randomizes them to prevent session hijacking and other attacks)
• Data offset is likely untouched unless traffic flows through sofisticated firewall
• Most of ICMP types and codes are blocked by typical firewall except Echo request|reply, destination unreachable and TTL expired in transit useful for traceroute
• Payload of ICMP echo and SEQ are likely untouched.
• Payload of TCP is typically untouched by firewall excepting some tricky cases (ALG gateways which modifies FTP, SIP etc, or HTTP/FTP proxies). But most of ports are blocked by corporate firewalls and others forwarded to proxies.
• TCP flags are typically untouched but TCP session follows some rules (handshake etc), so firewall (stateful) watches them and drops invalid or unrelated packets
• DF bit in IPv4 packet is typically untouched, probably other fields such as length and offset, but the packet should be of right size to be passed through
• QoS bits can be modified or completely zeroed traveling the public network.

Well, your question is the reverse-engineering research in a way. Appreciate you share your experience afterward.