How long is the $20 discount valid for (offered in partnership through my bank)? Do I need to file before a certain date to take advantage of this offer?

The offer is "up to" $20 off and it would depend on which product you are using.  You will need to access the TurboTax offer via your bank's website and start your return through that link (also go there each time you work on it), then finish and file from that same link.  You will need to file by February 19, 2018.