Do miners receive a reward when solving the hash for the next block or when validating the block, or both?
Do miners check if this transaction is valid (check if I have enough funds)?


Every node that receives the transaction will validate it, this includes the nodes miners run.


  If they do, do they receive a reward for this?


There is no reward for this


  My question is, if the block is invalid and gets sent to the blockchain, why other miners check if said block is valid?


They check it precisely to see if it is invalid. If a broadcasted block is invalid, and a miner starts mining the next block using the invalid block as their previous block, the next block mined by them will also be invalid, since it builds upon an invalid block.

This results in them losing any work invested in finding this new block, since it will be rejected by the network.


  Does the first miner that "solved" the block wrong get a reward?


If the block is invalid (“wrong”), then it will not be added to the blockchain by any other nodes/miners. So they will not get a reward. 

For finding a valid block, they receive a certain amount of new Bitcoin + transaction fees (in the coinbase transaction).


  Why would other miners check if it's correct when the miner already received the reward.


If they don't validate it, and it turns out to be incorrect, even if they succeed in finding the next block, they will receive nothing in return since it will be rejected for using an invalid block as the previous block.


  What do they get from checking the solved block if they are already working on the next one?


A guarantee that if they solve the next block, it will not be rejected for building on an invalid block.