1. Trying (guessing) to find a hash, which would fit as an answer to Bitcoin network for the current block. Each block needs a different answer and it varies all the time depending on how easy or hard it is to find that answer for miners.
Not really. Miners are hashing block header and changing it after every hash to try to meet the target[1] by increasing the nonce. The block header contains various information such as the version number, hash of previous block, merkle root, timestamp etc.
Once the block fits the target requirement, it is considered valid by the network. Hence, every block within the same difficulty period are equally difficult to find.
2. Verifying transcations, made by other bitcoin users at the moment (while this block is being solved) and writing the transcation data to the block.
It is technically not writing transaction data into the group. The hash of the transactions are all hashed in the merkle root so the transaction is technically already in the block when the miner starts on the block.
Every single node in the network verifies the transaction, the miners are merely putting it into the blockchain.
Of course, there are other things involved into mining, as providing security, writing the data of a previous block,
They don't mess with the previous block. They just expand on the previous block by including the hash of the previous block.
however I basically talk about the Reward part, how miners are earning cash. In the first case, whenever a block is solved (which can be done only by one person at the time), the miner gets a reward, which currently is 12 BTC. There are pools, who are looking for correct hash together and then splitting the profits afterwards. So, in this case, the hash speed is the most crutial part, which is needed to look for the correct hash to solve the current block.
The miner gets 12.5BTC+transaction fees that is from the transactions in the block.
However, I would like to know more about the second part. Miners also earn money by verifying transactions, right?
They do verify the transaction but it is not the main reason why they are rewarded.
-snip-
I think you misunderstood the entire mining process so I'll just explain how it works.
Firstly, the miner assembles a set of transaction that he wishes to include in his block. The merkle root should contain the transactions that the miner is planning to include, the coinbase transaction which contains the 12.5BTC reward and all the mining fee that is paid by the transaction in the block. Next, the miner combines this with other information such as the version number, hash of previous block, merkle root, nonce, etc and this is the block header. Next, the miner starts to hash the block header till it reaches the target and if it does, the miner relays it to the network. The nodes in the network will examine the block and choose to accept or not to accept the block according to their own rules, eg. minimum difficulty validity of transactions.
If the block doesn't hit the target, the miner will change the nonce or the information in the transaction to try again. The faster you can calculate hashes, the higher your probability of hashing a block header that can meet the target.
[1]
https://en.bitcoin.it/wiki/Target