Bit offtopic, but I highly recommend completely ignoring consumer WiFi gear these days. Enterprise grade APs are getting pretty cheap these days, and the difference is night and day. Open Mesh used to be a lot cheaper, but even at $80 (half that for obsolete models on eBay), it's hilarious to compare them to any consumer off the shelf device. Unless you have a very large house or walls made out of concrete/brick/steel, a single Ubiquiti access point will cover your entire house.
Also, generally do not recommend using any WiFi built into your modem. Separate hardware is better. Keep it up to date with the latest firmware. One reason for my annoyance at consumer hardware is that they tend to be the majority of botnets these days and get infected all the bloody time. When your WiFi is no longer supported by the manufacturer, toss it and get another.
End of rant. To answer your questions:
1) Depends on model, but either is possible. With consumer gear, you're better off coding the extender to your DSL Modem's SSID (fancy term for wifi network) and then having a separate SSID for your new wireless network.
2) Depends on model, but yes, it is possible to do so. No code is necessary because ethernet doesn't typically come with passwords.
Remember, your speed drops in half for every hop you make.