What approach to take in modifying a 8051 based board?
Is there a way to get a dump of the current hex code that is on the chip? After messing with my own code, it would be a good fail safe to have so I could always revert to the original code.


Only SinoWealth or someone experienced with that specific product family will be able to answer that. 


  Can I use any 8051 toolchain to work with this specific IC, or do I need to use whatever SinoWealth is offering?


Any 8051-targetting C compiler will convert C to valid machine code for that MCU.

But that doesn't mean that addresses of memory segments are right (i.e. you need linker scripts that match your hardware), or that any of the peripherals work like in any other 8051 MCU.

So, that's a "yes, with extensive work in making sure to tell the toolchain about the properties of the chip you're working with, and then potentially writing drivers for the more complex peripherals". 


  Someone suggested certain Atmega's are pin compatible with the 8051 but I cannot find much info on that. Is it true?


Why would that matter? What you're looking at is not an original Intel MCS-51, but a microcontroller that has the same CPU core; it's more than questionable your low-cost clone should have the same pinout as a 1980 Intel microprocessor.


  As an alternative to reprogramming the 8051, I thought to pull down the RST pin and use another IC instead. However the 8051 doesn't reset when the RST pin is pulled down, however it does crash until I cycle the power when pulled high.
  
  &nbsp; Why doesn't it stop functioning when pulled low and would it be safe to keep it high?


I'd agree, simply investing cents to euros to get a modern MCU in there (e.g. one of the cheaper STM32) would make much more sense – especially since programming hardware that you'll need to flash your programs to the 8051-clone will not be free, either.

Why not simply measure all the DC voltages at all the 44 pins (not that many), figure out the power supply and ground pins, and use a cheap hot air gun to unsolder the MCU alltogether and use a cheap PCB board service (e.g. OSHpark) to manufacture an adapter board that allows you to use ribbon cables to connect your replacement to the original MCU's pads?

If that pays at all - often, it's easier to just scrap the control board alltogether and just interface with the actors (and if any, sensors) of the device directly from an MCU eval board of choice – these can be had for 2 to 20 €.