I set the SSID for the 5Ghz different from 2.4Ghz
Google WiFi does everything on a single SSID and it's completely seamless. I never worry about one vs. the other and all our devices magically switch to whichever frequency is appropriate. I suspect there's some active management going on in the APs to make this work.
Agree with running CAT6 cables to the equipment closet locations. Run more than one to each location. Cables do go bad sometimes, sometimes well after installation.
I endorse running extra wires to each location. Better to have wires in place that you don't need than to eventually discover a need for wires that you don't have. To that end, there are a number of cable products that carry multiple Ethernet and RG-6 cable TV wires in a single bundle (
example bulk cable from Amazon). If I was running wires from scratch, this is what I'd use.
I don't think it particularly matters whether you're using Cat5e, Cat6, or newer. Cat5e is all you need to run gigabit Ethernet, and supposedly the higher-rated cables are very much more particular in terms of minimum bend radius and such.
A good wired connection provides the best reliability. WiFi can be quite good, but sometimes the solid connection using wired Ethernet is worth the hassle of installing it.
Without any doubt, a wired "backhaul" network for your APs is the preferable solution whenever possible.