Dual-band wifi router to dual-band extender: can/should I use a single SSID name?

Solution 1:

You can certainly share the same ESSID between multiple access points, as long as:

• they also have identical security settings (e.g. WPA2-PSK with AES);
• they connect to the same IP subnet (otherwise roaming devices in overlapping areas can get very confused).

Beyond that, no special configuration is needed – clients will roam to whichever AP they prefer.

Performance problems are mainly caused by extenders' uplink quality and radio signal interference. For example, a pure repeater which shares the same radio & antenna for everything will perform much worse than one with dedicated uplink radio (especially since it can use a different channel/frequency), which will still perform worse than a wired access point.

The SSID doesn't affect this – besides not being able to manually choose which AP to use anymore. (Sometimes the client might insist on using a slow repeater due to stronger signal.) Commercial "mesh WiFi" products have features to steer the client towards a faster AP.