Does this support wireless mouse and keyboard?

No it does not. You can connect a multi device mouse and keyboard though. That's the setup I have: Logitech keyboard and a Microsoft bluetooth mouse.

My laptop allows USB-C for the monitor. Can I use that with this or does it require me to connect HDMI from my laptop?

I could not get this working. I don't believe it's supported. It has 2 HDMI-in ports, I believe it's limited to only HDMI-in = HDMI-out

Questions & Answers: 2 Port Dual Monitor 4K HDMI KVM Switch, 2x2 HDMI USB 3.0 KVM Switch 2 in 2 out 4K @60Hz HDMI 2.0 Switcher for 2 PC Share Monitor Mouse Keyboard (with USB 3.0 port)

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2 Port Dual Monitor 4K HDMI KVM Switch, 2x2 HDMI USB 3.0 KVM Switch 2 in 2 out 4K @60Hz HDMI 2.0 Switcher for 2 PC Share Monitor Mouse Keyboard (with USB 3.0 port)

[HDMI KVM USB switch dual monitor] This KVM Switch allows control 2 computer with one set of controls (keyboard and mouse) and two display screen. Realizes 2 in and 2 out free selection synchronous switching. You can centrally control and manage hosts laptop or game console without having to move from one workstation to another.
[UHD 4K Visual Enjoyment] The kvm switch supports HDMI 2.0 and standard HDCP 2.2. Support max resolution up to 4K@60Hz,1080@144Hz, also support 1080P@120Hz or lower resolution
[Dual HDMI KVM Switch with USB 3.0 Port]:Supports Connecting your USB 3.0 port device such as Scanner/USB driver/3D Printer/Monitor Camera
[2.0CH Audio Extractor] KVM Switch comes with a 3.5mm Audio Jack port with a DAC chip inside, which can be inserted into a 3-segment 3.5mm earpiece device.
[Wide Compatibility] Support mirror mode,extend mode and crisscross mode Compatible with all laptops with HDMI port and other devices.It has adaptive EDID, adapt to Windows, m-a-c OS, Linux, UNIX...

$68^.99 –