No
This is a Monster Spawner. The monster will spin faster and faster and eventually spawn that monster nearby. Note that if you are on Peaceful setting, the monster will not be spawned. You can destroy them, or you can adjust the area around it to 'farm' the creature, ie, let it be created, killed by the environment, and harvest the drop item. The spawner can be temporarily disabled by putting a torch right next to it, while you work.
The mob spawner cage's official item number is: 52
Minecraft's Item ID's can be found on Minecraft Wiki. Link in related links.
No, you can't take one item from a Minecraft mod unless there is a separate mod which contains that one item.
In Minecraft, you cannot obtain a spawn cage directly as a player item; spawn cages are part of the game’s structure for mob spawning. However, you can create a similar effect by using a command block to generate a specific mob spawner. To do this, use the command /setblock ~ ~ ~ minecraft:spawner{SpawnData:{id:"minecraft:[mob_name]"}}, replacing [mob_name] with the desired mob type. Make sure you have cheats enabled to use command blocks.
The item ID for a diamond pick in Minecraft is 278. I have linked you below in the related links of a full list of all the ID's in Minecraft.
Minecraft
46
You can't.
there is no "carpet" item/block in minecraft, but people use different colored wool as carpet for minecraft.
There are no quarry blocks in Minecraft. Those are added by Buildcraft, a mod.
you right click on a block