Luckily, it looks like the water damage is contained to that small area and shows typical intrusion problems. Thank your lucky stars that the water hasn't completely done in a whole wall or roof.
The pieces you need to access for the corner repair will need the siding detached on the exterior. The usual process for something like this would be: Remove vinyl screw cover in aluminum corner trim rail. Remove screws holding trim rail onto the corner framing. This should expose the ends of the aluminum siding. You will see many staples and maybe a screw or two holding the siding to the wood frame. Remove all staples and other fasteners. When you have this much loose, you'll be able to decide if pulling the vertical corner pieces is doable or if you have to go back to an adjoining stud and remove more fasteners to have room. Done carefully, the former might be all that is necessary. Cut out the corner vertical frame at the bottom and top plates and any horizontal frame members (there is usually one across the front).
Replace with adequate frame lumber. I used pressure treated wood and cut odd sized pieces on a table saw. Installation of the new pieces is done with construction adhesive on the joints that mate to the top and bottom plate, also to the horizontal frame. I used nail guns and screws to toenail everything back together. Make sure you observe the original dimensions so that everything fits back together easily. I used an elastomeric sealant between the siding and frame pieces before reattaching. Two beads of the same sealant were used under the aluminum corner trim. Refasten siding and aluminum trim. Replace screw cover in aluminum trim rail.
If you have a corner trim that is not like my old school rig but has a molded plastic corner, it may be necessary to detach siding further back from the corner and hunt out those fasteners that hold the corner trim to the frame. On some, the corner trim is installed first and the siding laid into the corner. In either of the two situations above, the corner trim should come off easily as the frame has given up a lot of holding power.
Yes, you will have to detach the interior panel in the affected area. If you can find the joint in the paneling and work carefully with small pry bars, the panel may be saved. Most of the time, the interior panel is just stapled to the studs but some mfgs may use glue, too. If you can't salvage the panel, that's just how it will have to be. Replacement Lauan paneling is still less than $10/4'x8' sheet.
The floor: I haven't seen a TT floor, yet that is not full width to reach under the wall. If you have some rotten floor in just one area and your floor is a single sheet subfloor (5/8" OSB or plywood), the rotted floor sheathing can be worked out from under the wall and a new piece slid under the wall. There will be fasteners in your way and they will need to be cut with a reciprocating saw or grinder. Again, glue and screw the new piece onto the frame.
If you have spun fiberglass batt insulation, I'd suggest that you replace it with extruded polystyrene foam board. The foam board can be cut and friction fit into place and will afford better R-value and stay dryer.
After you have the frame replacement done, we can share more on finding leaks, Id be happy to tell you how I test for that and my experience with leaks.
