Do they normally have dedicated eco system being peer-to-peer payment system, have their server pools, like Bitcoin network? Or these "coins" are some kind abstractions, "tokens" which just can be traded on crypto exchanges?
Doge have it but it's not for shiba. Shiba was ERC20 asset and this asset pegged with ethereum blockchain. That's why shiba was different with doge and bitcoin that used its own network.
How do the new cryptos which work like bitcoin with their network and proof-of-work, protect from an attack of someone who has significantly higher CPU power (pool of servers), who can re-do the proof-of-work rebuilding the block chain in the attacker favor?
The majority of new cryptos are using POS instead of POW due to the efficientcy. I think that you must read some articles about POS. This is a proof of stake mechanism that lets stakers to hodl the blockchain.
Do the first coins appear from mining first blocks, or there are first transactions with some kind of initial deposits?
That depends on what consensus that used by coin. If that was using POW and it will be appearing from mining the first block but the different implementation when it comes to the POS