The private key is encrypted to generate a digital signature,
No it is not.
Private key is used in a mathematical equation to create the signature using the message hash.
The function of digital signature is to verify that data is not modified during transmission.
Not exactly correct.
Checksum (hash) is used for integrity check (ie. make sure data was not modified
during transmission) DSA is used for authenticity check (ie. make sure the intact data that was downloaded is coming from the source we expect it came from) this process covers integrity check too.
Digital signature is an upgrade of one-way encryption.
Wrong. DSA is used mainly for authentication not encryption.
The quote here seems out of context. I also didn't get what certain parts are like "q" that is said to be 160 bits!
I am not a native English speaker.
It is not your English, there are fundamental flaws in what you explain.