The more transactions are done, the more work will be needed to validate it
No. The validation time remains the same, regardless of how many transactions. It's about 10 minutes give or take.
To compensate for the work needed, a small amount of bitcoin is given to the one who validate the transaction
You seem to be confusing blocks (a group of transactions) with transactions. A transaction just takes a few milliseconds to validate. The bitcoin node can look at it (not even a miner) and verify that it has valid op_codes, it falls within current currency ranges, and that it's coming from a good node, among other things. There is no reward for validating a transaction. Solving a block, however, comes with a reward.
But when we will continue, and with the more widespread use of Bitcoins, we'll reach a time when transactions will takes days
This is 100% incorrect. Transactions will always be nearly instantaneous. If there are more transactions than the network can handle we could innovate in new ways, like using side chains.