This seems to be one of those Logitech universal receivers. I'm not too familiar with these, but let's look at relevant parts from dmesg:
[ 1.489700] logitech-djreceiver 0003:046D:C52B.0003: hiddev0,hidraw0: USB HID v1.11 Device [Logitech USB Receiver] on usb-0000:00:14.0-13/input2
The driver itself registers as HID-only.
[ 1.617557] logitech-hidpp-device 0003:046D:4101.0004: hidraw1: USB HID v1.11 Keyboard [Logitech T650] on usb-0000:00:14.0-13:1
This seems to be a wireless touchpad, which is recognized by the logitech driver, but again registers HID-only.
[ 2.066592] input: Logitech Logitech BT Mini-Receiver as /devices/pci0000:00/0000:00:14.0/usb3/3-14/3-14.2/3-14.2:1.0/0003:046D:C71B.0006/input/input2 [ 2.123659] hid-generic 0003:046D:C71B.0006: input,hidraw2: USB HID v1.11 Keyboard [Logitech Logitech BT Mini-Receiver] on usb-0000:00:14.0-14.2/input0 [ 2.123833] input: Logitech Logitech BT Mini-Receiver as /devices/pci0000:00/0000:00:14.0/usb3/3-14/3-14.3/3-14.3:1.0/0003:046D:C71C.0007/input/input3 [ 2.183786] hid-generic 0003:046D:C71C.0007: input,hiddev0,hidraw3: USB HID v1.11 Mouse [Logitech Logitech BT Mini-Receiver] on usb-0000:00:14.0-14.3/input0
The Logitech driver registers at the input layer, with two generic input event endpoints.
[ 6.625441] input: Logitech Performance MX as /devices/pci0000:00/0000:00:14.0/usb3/3-13/3-13:1.2/0003:046D:C52B.0003/0003:046D:101A.0005/input/input4 [ 6.625594] logitech-hidpp-device 0003:046D:101A.0005: input,hidraw4: USB HID v1.11 Mouse [Logitech Performance MX] on usb-0000:00:14.0-13:2
The Logitech Performance MX mouse is detected by the Logitech driver, and registered both as HID-device, and with the input layer.
What does not happen is that any Logitech keyboard is detected by the Logitech driver. What also does not happen is that any driver for PS/2 controller shows up (usually as PNP0303, PNP030B or similar).
The computer seems to be an Asus EEE Notebook.
Diagnosis so far:
1) PS/2 and BT problems are unrelated.
2) The PS/2 controller hardware may be disabled in the BIOS. It may also need fiddling with ACPI options to work on this computer, or it may even need a special driver. This will need trial and error, googling and a bit of work to get right. Booting into Windows if possible, and finding out the I/O ports for the PS/2 controller could help.
3) Either the keyboard and BT receiver are not paired properly, or something else causes them to not work together. When you tested the BT keyboard on another computer, did you also test this BT receiver? Was the other computer a Linux computer? If yes and it works, can you compare the dmesg from this computer?
Edit
Ok. standard i8042 PS/2 controller seems only to get enabled when something is plugged in:
[ 0.737951] i8042: PNP: PS/2 Controller [PNP0303:PS2K] at 0x60,0x64 irq 1 [ 0.737952] i8042: PNP: PS/2 appears to have AUX port disabled, if this is incorrect please boot with i8042.nopnp [ 0.738577] serio: i8042 KBD port at 0x60,0x64 irq 1 [ 0.738783] mousedev: PS/2 mouse device common for all mice
No change for the Logitech driver. See above for what to try. If you have a documentation about for the Logitech (how to pair etc.), that may also help.
dmesgafter boot to see if there are any errors when the keyboard drivers try to initialize. Second step is to find out which layer fails: USB (usbmon), input (evtest), or X (xev). As both PS/2 and BT/USB keyboards don't work, I suspect trouble with the kernel keymaps.dmesgoutput. (output was too long to fit in post)