How to be assured a wallet is on a blockchain? How could I check that a wallet or any app I download from Google play or Apple store or a website is truely runs by a blockchain and be certain that producing company can not manipulate the private key?
That's impossible. The only way you can be sure is that you compile the wallet from the source code by yourself. In practice, you should only use wallets that are famous (some of those are Blockchain.info wallet, Bitcoin.com, CoPay, Coinbase, Electrum, BRD,... I can count at least 20 of them). You can search online with the name of the wallet software you want to download and refrain from downloading it if there isn't enough of convincing reviews.