Should 5GHz and 2.4GHz use same SSID?

Hello all,

Hope I'm in right place to find help I need.

I have 5GHz router from my ISP but it does not cover my whole house so I have used my old D-Link DIR-655 and set it in bridge mode so it works as AP. Unfortunately this router does not have 5GHz mode so it is being used in 2.4 GHz mode.

Now few things:

1. I have issues with setting right channel for D-link as I have many neighbors and that affects speed a lot but most of the time it is fine for my use. But I have noticed that the modern devices don't switch to 5GHz even if I am next to main router. I have to disconnect and connect to wi-fi and then 5GHz is chosen otherwise devices stay on 2.4GHz connection.
2. knowing above should both routers use same SSID/password? That is my setup atm. I have no idea if setting up 2 different SSID/pass makes things any different? Will phones/laptops that can use 5Ghz prioritize that connection when in range of both routers?