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When a create a wallet, lets say with blockchain, do I have to check my wallet always with blockchain, or can I also check my wallet with different providern of wallets, where I did not established my wallet. Some provider of wallets link directly to blockchain, and some not. Therefore I could a bit confused.
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No, you can import your private key in another wallet software. Under the import/export option in Blockchain you can export the private key, and you'll be able to import elsewhere. It also works the other way around. If you create an address somewhere else, you can also import it on the blockchain website.
So no, you do not always have to check your wallet with blockchain.
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With Bitcoin and cryptocurrencies, you're free to do everything you want with your money. The private key shows ownership and any wallet software that can import your private key will give you full control (i.e. issue transactions).
Now, if you just want to see the and monitor transactions related to your address, this is public information that can be seen on any block explorer. For example:
https://helloblock.io/addresses/18DUBayAjhP4Tbf3hvkSmyMuvbHB2L8siy
