The Secp256k1 curve is used in cryptocurrency. Can someone generate a private key with a different curve? Many cryptocurrencies use Secp256k1. 

Every cryptocurrency library comes with its own redundant implementation of Secp256k1, ECDSA, RIPEMD160, and SHA256. So, there can be some inconsistencies across implementations.

What if some library generates a private key using a different curve than Secp256k1? For example, one with less security, or even a backdoor. Choosing different points, etc.

Then they share only the public key with blockchain, of course. Is it possible that such a thing would work? Can some public key appear valid for Secp256k1, but in fact use another elliptic curve?

I just came across many open source cryptocurrency libraries where they generate private keys and everyone is using custom implementations of these. So I should at least compare source code and look for some specific set of constants.
The Bitcoin network expects secp256k1 when adding something to its blockchain. If you use another curve, it naturally will not produce a valid secp256k1 public key capable of validating the secp256k1 signatures and will be rejected by the network. If some client uses a bugged or even backdoored version of the curve, the client will simply not be able to add anything to the blockchain. This would only become a risk if a significant portion of the network ended up using this same bad library, effectively resulting in a fork of the core client and an entirely new, separate protocol.