As for the original Optimus Prime...
From his original tech specs:
"Splits into three autonomous modules: 1) Optimus Prime... the brain centre known as the Commander; 2) Roller, the Autobot scout car... a spy who operates up to 1931km away; and 3) Autobot Headquarters... the combat deck equipped with a versatile mechanic/artillery robot. Injury to one is felt by the other two."
From the G1 TFU profile:
"The Optimus module is the robotic sentient being that is the storehouse of his vast knowledge and strength."
"...the Optimus module is by far the most important component of the trio. Although he could survive without the other two, they could not survive without him."
It is possible to still form a gestalt with a dead component, although the gestalt robot itself suffers from reduced physical attributes like speed, dexterity, mobility etc.
Some gestalt teams do have "spare" members who are already designed to combine with the team and can replace a member, e.g. Deathcobra.
They also added personal reinforcements with their remote control. The downside is that any damage dealt to the shell is felt by its owner, but only even more intensely. Destroying the shell will make its owner pass out. As for disguise... this occurred more in Japanese continuity where the Pretenders are able to mass-shift and shrink themselves down to human size. In Anglophone continuity Pretenders never had this ability and so we rarely saw Pretenders using their shells as robots in disguise unless they happened to be on a world inhabited by giant humans (e.g. Femax, Pz'Zazz).
I'd say it'd be much like Optimus Prime -- whichever component houses the "brain" would be the most pivotal... so my guess is the head component. So Battletrap could survive if the camper truck component were destroyed, but not if the helicopter were destroyed. Likewise Flywheels could survive as only the jet, but the tank would also perish if the jet were destroyed etc.
Sky Lynx is a trickier one because both components have heads and there's a lack of canonical evidence to suggest anything.
Old?? In the G1 comics Omega Supreme was constructed on Earth by Grapple - making him one of the younger Autobots. In the G1 cartoon he was created on Cybertron, but so were most Autobots and Decepticons. Was it because he was around before the war started? So were quite a few Transformers, including Optimus Prime (as Orion Pax). Also, the concept of "reformatting" or "transcanning" are new concepts that didn't exist in G1. Transformers were physically reconstructed to have new alt modes, such as when Auntie and Teletraan I rebuilt the occupants of the Ark. Either way, I don't see any canonical evidence that suggests that Omega Supreme couldn't handle being rebuilt with a new alt mode. He's not like say, Emirate Xaaron - where canon has explicitly stated that he has become physically unable to transform into his tank mode ever again (and attempting to do so may kill him.
I'd say that Omega Supreme can survive so long as his head's in tact. Destroy the rocket? He becomes an amputee, but hardly fatal. Unless you wanna tell me that destroying one of his little yellow clips is gonna kill him.
I don't think it was explicitly stated that Emirate Xaaron transformed into a tank, though I'm not against it. Most of Xaaron's body was modeled after Megatron's character model, though given that it was the 80's when this came out he literally could have had anything as his alternate mode, the only clue given is that his alternate mode would have been powerful enough to defeat Flame, and was some sort of assault vehicle, this doesn't leave out the possibility of an aircraft, or given his physical resemblance to Megatron, some sort of cannon, or gun. I've been debating this quite a bit myself recently since one of my future projects is an Emirate Xaaron custom, and I want him to have an alt mode.