You wire them to an PLC IO module.
Or, you get the addressable variant, like Eaton SmartWire, and talk to those.
You can even get wireless ones today!
What you should not do is wire the low voltage mcu signals directly to these buttons over cables. The cables and buttons are too susceptible to noise to be reliable that way. They also bounce a lot.
This type of pushbutton is designed to be used in circuits with higher voltages and currents than you'll use on your MCU, for example with motor contactors and machine tool relays. This is why they have screw terminals designed for heavier gauge wire than you need with your MCU.
If you must interconnect these to a PCB, your best bet is to have a set of PCB-...
It means that its tested at those specs: they pass at least 0.5A on all conductors and the surface temperature of the cable cannot rise more than 10C above ambient(23C in that case). That alone doesn't mean that you can't pass more than 0.5A, it just means that if you pass more, its not guaranteed to meet that spec.
I did not see an absolute maximum current ...
The hot plug detect pin is an output from the monitor.
You can freely ignore the pin, the monitor does not care.
However then you can't know when a monitor is connected or powered up and you can't know when it is a good time to read what video formats the monitor supports. Which basically means you can run into compatibility issues.
Just as a warning, a DVI-...
I expect that the brown wire is the Neutral (non-Hot) wire - not required for a switch, but would be required if you wanted to put an outlet in that box.
However, a web search suggests that Brown should be Hot 230 V and Blue should be Neutral.
It appears to be the third wire needed for a three-way switch setup (where a fixture can be controlled from two different switches. There may have been a three-way switch previously wired in that box and the third wire was never removed.
The voltage may be induced voltage from running parallel to live wires.
It looks like you have to push on the locking tab at the top wile pulling the connector straight out. See photo below. It may be a tight fit, I would use a large flat screwdriver (green arrow) to carefully prize the plug from the socket.
It looks like an extender cable for a gamecube controller.
See for example