Trying to understand the mempool and how transactions are picked I am very new to blockchain and try to understand it completely. What I think I understood is,
that every transaction is going to the mempool first and these transactions are getting picked by miners. With a bunch of transactions and the previous block hash a new hash needs to be calculated. And as soon as the desired hash is calculated a new block is mined.
What I dont understand:

Is one transaction only picked by one miner or is it possible, that more miners can pick the same transaction?
If only one miner can pick a transaction, what happens if this miner has weak hardware?
If more miners can pick a transaction, what happens to the other miners(that couldnt calculate the hash first).

I googled a lot and couldnt find an answer to this questions.
Hope this question meets the criteria and you guys can help me :).
Thanks in advance.
Solo Miners

Is one transaction only picked by one miner or is it possible, that more miners can pick the same transaction?

Each miner is free to pick whichever transactions they want. They have no knowledge of what other miners are working on. Miners might pick different transactions, some the same and some different or all the same. There is no explicit coordination.
Only when the rate of new transactions is low, it is likely that each miner will have the same set, or nearly the same set, of transactions to work with.

If only one miner can pick a transaction, [...]

Not applicable.

If more miners can pick a transaction, what happens to the other miners(that couldnt calculate the hash first).

When they see another miner has produced a new block, the others stop their work and start on the next block based on the new block produced by the other miner. They first remove from their mempool any transactions included in the other miner's new block.

Mining Pools
Things are a little more complicated for mining pools. Members of a pool coordinate their work cooperatively. But you can think of each separate pool as a separate miner and the above applies.
Stratum v1: Pools and solo miners are the only entities that are constructing block templates. Regular miners cannot because they don’t have the transaction sets to build the blocks.
Stratum v2: Every miner (who has it implemented in their firmware &amp; a pool that supports it) will have the option to prioritize transactions however they want without impacting the other miners in the pool.

Further reading

Slides about basics of Bitcoin Mining by Daniela Brozzoni
What exactly is Mining?
What are bitcoin miners really solving?