Minecraft – Test if player has specific item in specific hotbar slot [1.8]

This answer was written prior to the release of 1.13 and "The Flattening" that came with it. An updated answer can be found here.

You're actually pretty close, and definitely on the right track. Instead of {Inventory:[{id:<block ID>,data:<data value>}]}, use {Inventory:[{id:<block ID>,Damage:<data value>s}]}. The s after the data value is important since it's stored as a short, not as an int. (As an aside, if it were stored as a byte, you would append a b instead.)

It's very difficult to see whether someone has an object anywhere in their inventory. That would take 36 command blocks, one for each slot! You can, however, take the object, then see if taking the object worked.

The Commands

• Set up a fast clock (but not as fast as a fill clock)
• One command block does /clear @p iron_ingot 0 1 (clears 1 iron ingot from the nearest player. You can choose which player, but it must be exactly one. Also, you must use the same selector throughout all these commands.)
• Run a comparator from that command block to another command block running this command: /scoreboard players add @p ironIngots 1 (Increases the player's ironIngots score by 1. Make sure you create ironIngots first!)
• Create another command block that does whatever you want. In the case of giving them a diamond, you would do this: /give @p[score_ironIngots_min=10] diamond. Note the target selector; it gives the player the diamond only if their ironIngots score is 10 or greater.
• The last thing you want to do is take the ironIngots score away so they don't keep getting items. Just run /scoreboard players remove @p[score_ironIngots_min=10] ironIngots 10. This removes 10 iron ingots from the player if they have an ironIngots score of 10 or greater. Any player who matches this will have received a diamond in the previous step.

How to Set Them Up

The commands listed above must run in order. It is actually very simple to put them in order. Just open the debug menu and look at the stylish new crosshair. Align the command blocks along either the green or red line so that one of those lines points down the first to the last command block.

Notes

• You don't have to press a button to use this, you just have to be the nearest player. You can change the command to the nearest player within 10 blocks, nearest player on a team, etc.
• The commands will take a short amount of time to finish, but for 10 iron ingots, it will only take half a second. It's one tick (1/20 of a second) per item taken away.
• It will take away iron ingots even if you don't have ten. However, you can submit one, then later submit five, then submit four more later, and it will still give you the diamond. You can also submit 14 at a time and the extra 4 will be saved for later.