Can the Homey gateway connect with the Tuya brand’s app or products?"

Yes, only for Homey Pro models, and only for Tuya Wifi & Zigbee based products, by these community apps:

  • Tuya Zigbee: [APP][Pro] Tuya Zigbee App

    • There’s a 50% chance supported zigbee devices actually get recognised, while Tuya is flaky concerning firmware & device ID’s
  • Tuya Cloud (Tuya wifi, and Tuya zigbee combined with a Tuya zigbee bridge):
    [App][Pro] Tuya Cloud
    This app is “as is”: unmaintained and the used Tuya API should not be changed by Tuya company