What kind of switches does this keyboard use, and is it backlit?

It uses Brown Switch mechanical keys and includes RGB backlighting.

Does this keyboard support recording macros directly on the keyboard without installing extra software?

Yes. It has 5 programmable macro keys (G1 to G5) that can record macros on the fly without needing any additional software.

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Redragon K580 VATA RGB LED Backlit Mechanical Gaming Keyboard with Macro Keys & Dedicated Media Controls, Onboard Macro Recording (Brown Switches)

5 Macro Keys: There are 5 programmable macro keys(G1~G5) on the keyboard which can be recorded macros on the fly without any additional software required to be installed.
Dedicated Multimedia Controls: The multimedia controls let you quickly play, pause, and skip the music right from the keyboard without interrupting your game. Also, designed with a volume/backlight adjust the wheel, it's easy to adjust volume or backlight brightness directly with the wheel.
N-key Rollover: 104 floating keys anti-ghosting allows you to simultaneously click multiple keys.
Durability: 50 million times keystroke test, 60g actuation force, and 2.3 mm short travel. Special Double-shot injection molded keycaps never fade key font with waterproof and dust resistant Redragon brown mechanical switches.
RGB Backlight: 18 backlight models allow you to type in the dark. You can adjust its brightness with a control wheel or FN + Up/Down. 5 modes of RGB side edge lighting.