Depending on your generation, the rod of ropes (Complete Scoundrel 116) (4,000 gp; 4 lbs.) is either a Bat-claw, a hookshot, or both, but Dungeons and Dragons 3.5's attempt to simulate this complex mechanism with its rod of ropes was not without its issues.
The rod's Effect says that "no two functions can be used at the same time." If a wielder presses button one once to extrude 60 ft. of rope then presses button one a second time to stop the rope from extruding then presses button two once to activate the loops from one end and the hooks from the other, what happens? Pick a different scenario if its easier; in other words must the same button be pressed thrice before a different button can be pressed? (Note: I'm familiar with the hero-system, so you can reference a multipower if you want.)
The description of the rope effect says that the "rope has the same hardness and hit points as the rest of the rod." But this rod isn't given hardness and hp. The Dungeon Master's Guide on Rods says
Rods weigh approximately 5 pounds. They range from 2 feet to 3 feet long and are usually made of iron or some other metal. [...] These sturdy items have AC 9, 10 hit points, hardness 10, and a break DC of 27. (233)
Yet Complete Scoundrel describes the rod of ropes as specifically made "from the darkest teak and repeatedly stained until it holds a lustrous black shine." (And, if it makes a difference, the rod of ropes weighs 4 lbs. instead of the 5 lbs. of the typical magic rod.) Should the DMG's hardness and hp for a magic rod be used anyway, or are there better or more accurate statistics?
If the rope the rod extrudes is severed, what happens to that length of rope? What happens to the rod?
Nothing happens. The second function can't be used until the first function is deactivated, per the rod's description to that effect.
Specific beats general: it has the physical stats for rods as in the DMG, with the alteration that it's 4 lbs. instead of 5 lbs. Rods in the DMG have all kinds of materials but share break DCs, etc., so think of it as an inherent feature of “magic rodness” that supersedes the original materials' properties.
Don't break your magic items! The rope is an integral part of the item and severing the rope is breakage. The rod of ropes will be destroyed by such breakage, as with any other magic item when it breaks (excepting the exceptions that explode as well, of course). Per (2) though, it's very resilient rope compared to the mundane stuff, so accidental severing isn't likely and even deliberate severing will be somewhat challenging.
I completely disagree with the answer given to question 3. I would argue that the rod is a beefed up version of a spool of endless rope. Any portion of rope cut from the rod vanishes instantly but is retained by the rod.
2nd, to the second question, I would say that the rod's composition only matters for certain spells or attacks, for example heat/chill metal or a rust monster's attack wouldn't affect it but repel/warp wood would. It's a wooden rod after all. Sunder the rod and it breaks, but cut the rope and the user may fall but still has a functioning rod.