Hi,
I have experience for 15+ years in electronics/embedded development. I have several years of advanced AVR/arduino development, including library development from scratch. Practically all Arduino AVR boards used, like: 2009, 2010, UNO, MEGA, NANO, MICRO, MINI, etc. Several shields used like Ethernet (W5100, ENC28J60, etc.) with various IP protocols implementation, various graphical TFTs with and without touchscreen and various BUS types (parallel, SPI, I2C, etc.), all kinds of sensors and buses, serial and parallel memories, etc.
I have done a lot of projects with Arduino Mega and Uno boards with the standard ethernet shield (W5100 chip) implementing TCP (and UDP) clients interfacing with servers in intranets and internet, including your use case, a web server.
Controlling outputs based on precise (or not) timings is also no problem for me, both with blocking and non-blocking implementations, that really depends on your needs.
I have all the material needed for implementing your project for demo purposes before sending the complete sketch to you (I think I will only need an arduino and ethernet shield and some LEDs for visual feedback, but also have relays and relay boards if needed). Just some more details need to be discussed, like the number of outputs, timings, sequences, maybe access control, etc.
If you are interested in my bid, feel free to contact me.
Regards,
Carlos.