Is there any relation between closed source and open source wallet and only one of them is able to import the private keys or wallet with mnemonic phrases?
No.
Whether or not a wallet allow you to import a private key or seed phrase depends on how it has been designed by developers.
I am not sure what is wrong with that app but it's becoming very complicated over the time. Whenever I click on the New wallet addition it instantly creates new wallet, gives me new address and there is just no option at all where I can import external address.
To import a private key in mycelium, tap on 3 dot button at top right corner of the screen, select "Add account" and then tap on "Advanced & More".
After that, you will see options to import a single private key through copy-pasting or scanning a QR code.
It just don't show as the main one
That's what expected.
The address you want to sign message from has been already used and when you tap on "Receive" on bluewallet, it gives you another address. That's for protecting your privacy.
Take note that your HD wallet includes numerous addresses.